#72258173

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#1776
commutator	C₄×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₈²	C₂³
socle	C₂²	#8286
prime radical	C₂³	#1592
LCS	G ▷ C₄×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #11402 ◁ G
LECS	G ▷ C₂×C₈² ▷ C₄² ▷ C₂² ▷ C₁

296 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^10|a^2=d,b^2=dhi,c^2=degi,e^2=g,f^2=hij,g^2=i,h^2=j,b^a=bf,c^a=cfghi,c^b=cfg,d^b=dj,d^c=di,e^a=eh,e^b=eh,e^c=ei,f^a=fhij,f^b=fhij,f^c=fj,g^a=gj,g^b=gj,h^a=hj,h^b=hj

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72258182
maximal normal subgroup of	#72258182
maximal quotient of	#72258176

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