#72309674

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#442
commutator	C₂×C₈	C₂²×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
socle	C₂²	C₂×#1762
prime radical	C₂²×#135	C₂×C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×C₄×#97 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2096 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^10|a^2=e,b^2=f,c^2=efghi,e^2=h,f^2=hi,g^2=i,i^2=j,b^a=bg,c^b=cij,d^a=dhj,d^b=dh,e^b=ej,f^a=fi,f^c=fj,g^a=gi,g^c=gj,i^a=ij

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309687
maximal normal subgroup of	#72309687

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