#7623492

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623326
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

1348 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530688);