#72323180

order	2048
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#8146
commutator	C₂²×C₈	C₂×C₄×C₈
Frattini, cosocle	C₂×#1397	C₂³
socle	C₂	#72322700
prime radical	C₂×#41	#346
LCS	G ▷ C₂²×C₈ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂×#1397 ◁ G
LECS	G ▷ C₂×#1397 ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

561 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 128 points:

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

Polycyclic presentation:

2^11|a^2=d,b^2=hij,c^2=f,d^2=g,f^2=h,g^2=i,h^2=j,i^2=j,j^2=k,b^a=be,c^a=ceh,c^b=chij,d^c=dj,e^b=ek,f^a=fj,f^d=fk,g^c=gk,h^a=hk

Conjugacy classes & character table

212 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72323182
maximal normal subgroup of	#72323182

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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.8*F.9*F.10);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.5*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.11);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);