#72234417

order	8192
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₃₂² · Schur multiplier (H₂) = C₄×C₈²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²×C₈	C₈²
commutator	C₈	C₃₂²
Frattini, cosocle	#72234271	C₂²
socle, prime radical	C₂³	#72210714
LCS	G ▷ C₈ ▷ C₁
UCS	C₁ ◁ C₄²×C₈ ◁ G
LECS	G ▷ #72234271 ▷ C₄×C₈² ▷ C₂×C₄² ▷ C₂² ▷ C₁

1676 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 128
transitive degree	μ_t(G) = 8192

A minimal faithful permutation representation — 2 generators on 128 points:

(1,2,3,7,4,8,11,17,5,9,12,18,14,20,23,27,6,10,13,19,15,21,24,28,16,22,25,29,26,30,31,32)(65,66,68,73)(67,77,78,101,80,104,105,126)(69,75,84,95)(70,74,85,97)(71,76,83,96)(72,89,90,116,92,117,120,109)(79,103,106,119,108,91,118,93)(81,102,82,110,107,127,111,125)(86,99,113,94)(87,98,114,123)(88,100,112,122)(115,124,128,121);(33,34,35,39,36,40,43,49,37,41,44,50,46,52,55,59,38,42,45,51,47,53,56,60,48,54,57,61,58,62,63,64)(65,67)(66,72,74,91,76,92,99,103)(68,82,83,111)(69,79)(70,81)(71,80)(73,93,94,90,96,119,97,120)(75,77,98,102,100,104,124,127)(78,85,105,113)(84,89,112,117)(86,107)(87,109)(88,108)(95,110,121,101,122,125,123,126)(106,114,118,128)(115,116)

Polycyclic presentation:

2^13|a^2=c,b^2=d,c^2=f,d^2=g,e^2=hk,f^2=i,g^2=j,h^2=k,i^2=l,j^2=m,b^a=be,c^b=ch,d^a=dhk,d^c=dk,f^b=fk,g^a=gk

Conjugacy classes & character table

1472 conjugacy classes. The character table is too large to display.

Relationships

maximal quotient of

#72234420

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", "m");;
gap> coll := SingleCollector(F, [2, 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);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 8, F.11);;
gap> SetPower(coll, 9, F.12);;
gap> SetPower(coll, 10, F.13);;
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);;
gap> SetConjugate(coll, 4, 3, F.4*F.11);;
gap> SetConjugate(coll, 6, 2, F.6*F.11);;
gap> SetConjugate(coll, 7, 1, F.7*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);