#7623261

order	512
nilpotency class	4
isoclinism	isoclinic class of #7622979
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1409 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|c^2=gh,d^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,g^a=gh,h^a=hi,e^d=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322758
maximal normal subgroup of	#72322758

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530457);