#7624877

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624876
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	#62252	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1300 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)(9,10,14,18)(11,19,21,33)(12,24,26,39)(13,27,25,17)(15,28,29,22)(16,31,32,20)(23,38,37,35)(30,36,34,40);(1,3)(2,5)(4,8)(6,7);(9,11,20,36,13,23,26,28,14,21,31,40,25,37,12,22)(10,15,24,35,17,30,32,33,18,29,39,38,27,34,16,19);(3,8)(5,7);(2,6)(5,7)(11,21)(15,29)(19,33)(22,28)(23,37)(30,34)(35,38)(36,40)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532073);