#72321389

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#442
commutator	C₂×C₈	C₂³×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	C₂³×D₈	C₂×C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1811 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

(1,2)(3,4)(5,6,8,10,9,11,12,7)(13,14,17,24,19,25,20,35)(15,22,27,38,29,40,30,44)(16,23,18,34,33,41,31,21)(26,37,42,36,32,39,28,43);(14,21)(16,31)(17,20)(18,33)(22,36)(23,35)(24,41)(25,34)(26,28)(27,30)(32,42)(37,38)(39,44)(40,43);(13,15)(14,22)(16,32)(17,27)(18,28)(19,29)(20,30)(21,36)(23,39)(24,38)(25,40)(26,33)(31,42)(34,43)(35,44)(37,41);(2,4)(5,7,12,11,9,10,8,6)(13,16,20,31,19,33,17,18)(14,23,35,21,25,41,24,34)(15,26,30,28,29,32,27,42)(22,37,44,43,40,39,38,36)

Polycyclic presentation:

2^10|a^2=e,d^2=eh,e^2=h,g^2=i,i^2=j,b^a=bg,d^a=df,d^b=di,d^c=dj,e^b=ej,g^a=gi,g^b=gij,g^d=gj,i^a=ij,i^b=ij

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321463, #72321521, #72321666
maximal normal subgroup of	#72321463, #72321521, #72321666
maximal quotient of	#72322775

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