#7623222

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
prime radical	C₂×#41	C₂×D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1304 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=ghi,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530418);