#509136

order	512
nilpotency class	4
isoclinism	isoclinic class of #509065
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#13304	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

826 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ei,b^2=i,d^2=i,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,e^b=eh,h^b=hi,e^d=eh,h^d=hi

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, 416332);