#72308107

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
socle	C₂²	C₂²×#442
prime radical	#7622812	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ C₂³×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

4599 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^10|a^2=f,c^2=fi,d^2=fhi,f^2=j,h^2=i,i^2=j,b^a=bg,c^a=ci,d^a=dh,d^b=dg,d^c=dgij,e^d=ej,f^d=fj,h^a=hi,h^c=hj,h^d=hj,i^a=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308137, #72308232, #72308621
maximal normal subgroup of	#72308137, #72308232, #72308621

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