#72234406

order	4096
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₁₆² · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator	C₁₆	C₁₆²
Frattini, cosocle	#72234239	C₂²
socle	C₂	#72234265
prime radical	C₂³	#93995
LCS	G ▷ C₁₆ ▷ C₁
UCS	C₁ ◁ C₁₆ ◁ G
LECS	G ▷ #72234239 ▷ C₂×C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

575 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 256
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 2 generators on 256 points:

(1,2,4,14,7,17,35,74,9,18,37,76,52,86,141,200,11,19,39,77,56,90,139,205,61,92,36,147,53,87,142,201,12,20,40,78,57,91,146,199,60,93,148,75,51,168,140,204,10,62,38,149,50,167,145,247,8,58,59,171,6,16,34,73)(3,21,22,94,25,97,103,192,28,98,105,219,122,190,173,193,30,99,107,202,54,13,63,64,26,66,126,178,120,68,174,179,31,100,108,220,125,69,175,180,27,88,32,130,24,96,102,218,29,129,106,233,119,70,176,181,127,189,104,232,121,67,143,71)(5,41,42,151,44,153,157,250,46,15,79,80,158,82,206,209,48,83,33,132,123,213,134,244,165,84,136,72,43,152,156,195,49,154,137,196,55,81,170,208,128,85,135,197,159,89,133,215,47,166,138,210,162,211,144,194,45,163,164,198,160,216,245,203)(23,109,110,234,112,185,172,207,115,223,227,256,111,224,235,253,117,214,101,225,161,249,226,191,113,239,228,255,237,65,182,183,118,184,229,242,124,186,241,254,114,95,155,221,238,187,246,177,116,212,230,251,169,150,231,248,240,222,131,217,236,188,243,252);(1,3,5,23,8,26,45,113,10,29,47,116,60,28,128,115,12,31,49,118,61,127,165,240,11,30,48,117,9,27,46,114)(2,13,15,65,18,67,83,185,19,69,84,187,92,190,154,249,20,70,85,188,93,68,166,186,62,97,163,224,58,96,41,150)(4,32,33,131,36,108,135,230,38,126,42,155,37,107,136,229,40,105,138,228,59,22,79,101,39,104,137,227,148,106,164,110)(6,24,43,111,50,25,123,124,51,120,158,236,57,119,44,161,53,122,160,238,56,125,162,112,52,121,159,237,7,54,55,169)(14,71,72,191,75,192,151,183,77,193,197,254,171,64,132,177,78,179,198,248,76,180,196,252,149,218,80,207,147,181,210,253)(16,21,81,95,17,88,89,214,86,99,211,222,90,189,216,184,87,100,153,223,91,98,82,212,168,129,213,239,167,66,152,109)(34,63,133,172,139,173,157,243,140,103,156,231,35,143,144,246,142,176,206,241,145,102,170,182,141,175,245,226,146,174,134,235)(73,130,194,242,199,233,195,221,200,232,250,251,247,94,215,217,201,219,244,234,74,202,203,256,204,178,208,225,205,220,209,255)

Polycyclic presentation:

2^12|a^2=c,b^2=d,c^2=f,d^2=g,e^2=hkl,f^2=i,g^2=j,h^2=k,i^2=k,j^2=l,k^2=l,b^a=be,c^b=ch,d^a=dhkl,d^c=dkl,f^b=fk,f^d=fl,g^a=gkl,g^c=gl,i^b=il,j^a=jl

Conjugacy classes & character table

376 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234408
maximal normal subgroup of	#72234408
maximal quotient of	#72234407

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j", "k", "l");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.8*F.11*F.12);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 8, F.11);;
gap> SetPower(coll, 9, F.11);;
gap> SetPower(coll, 10, F.12);;
gap> SetPower(coll, 11, F.12);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.11*F.12);;
gap> SetConjugate(coll, 4, 3, F.4*F.11*F.12);;
gap> SetConjugate(coll, 6, 2, F.6*F.11);;
gap> SetConjugate(coll, 6, 4, F.6*F.12);;
gap> SetConjugate(coll, 7, 1, F.7*F.11*F.12);;
gap> SetConjugate(coll, 7, 3, F.7*F.12);;
gap> SetConjugate(coll, 9, 2, F.9*F.12);;
gap> SetConjugate(coll, 10, 1, F.10*F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);