#6374017

order	512
nilpotency class	3
isoclinism	isoclinic class of #3057
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	#60682	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1878 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6281213);