#72340992

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#20585
commutator	C₂³×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₂²⋊C₄	C₂⁴
prime radical	#9040967	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ #35348 ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

3891 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2,4,6)(3,9,11,16)(5,13,12,8)(7,14,15,10)(17,18,20,23)(19,22,25,29)(21,24,27,30)(26,28,31,32);(2,6)(3,10)(5,12)(7,16)(9,15)(11,14)(17,19,20,25)(18,22,23,29)(21,26,27,31)(24,28,30,32);(5,12)(8,13)(9,16)(10,14)(22,28)(29,32);(1,3,4,11)(2,7,6,15)(5,14,12,10)(8,16,13,9)(20,27)(22,28)

Polycyclic presentation:

2^10|a^2=e,b^2=ei,d^2=i,g^2=i,b^a=bi,c^a=cf,c^b=cfhj,d^a=dg,d^b=dg,e^d=eh,f^b=fj,f^d=fi,g^a=ghi,g^b=ghi,g^c=gi,g^d=gi,g^e=gj,h^a=hj,h^b=hj

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340995, #72341048
maximal normal subgroup of	#72341048

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