#7623327

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 ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1622 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)(3,7)(4,6)(5,8)(9,10,12,15)(11,20,21,17)(13,16,24,28)(14,27,25,19)(18,33,29,26)(22,34,37,30)(23,32,38,36)(31,39,35,40);(1,3)(2,5)(4,8)(6,7)(10,15)(16,28)(17,20)(18,29)(19,27)(30,34)(31,35)(32,36);(3,8)(5,7)(10,16)(13,26)(14,25)(15,28)(17,30)(18,19)(20,34)(22,39)(23,38)(24,33)(27,29)(31,32)(35,36)(37,40);(2,6)(5,7)(9,11)(10,17)(12,21)(13,22)(14,23)(15,20)(16,30)(18,31)(19,32)(24,37)(25,38)(26,39)(27,36)(28,34)(29,35)(33,40);(11,21)(17,20)(22,37)(23,38)(30,34)(31,35)(32,36)(39,40)

Polycyclic presentation:

2^9|a^2=i,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, 7530523);