#72237875

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#127215
commutator	C₂×C₂²⋊C₄	C₂×C₄²
Frattini, cosocle	#2308	C₂³
prime radical	C₂⁴×D₄	D₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂⁶ ◁ C₂×#2308 ◁ G
LECS	G ▷ #2308 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁

1861 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^10|a^2=f,b^2=d,d^2=j,e^2=hj,b^a=be,c^a=ch,c^b=ci,d^a=dg,e^a=ej,e^b=egh,e^d=eij,f^b=fh,f^d=fj,g^a=gj,g^b=gi,g^e=gj,h^b=hj,i^a=ij

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237877, #72237880, #72237926, …
maximal normal subgroup of	#72237877, #72237880, #72237926, …

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