#509144

order	512
nilpotency class	4
isoclinism	isoclinic class of #509081
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	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

810 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)(9,10,13,17,30,27,39,29,15,12,24,26,14,18,31,35)(11,19,21,36,32,20,38,25,23,37,40,33,22,34,28,16);(5,6)(7,8)(9,11,15,23)(10,16,12,25)(13,28,24,38)(14,32,30,22)(17,34,26,20)(18,36,27,33)(19,29,37,35)(21,39,40,31);(2,4)(6,8);(2,4)(9,12,15,10)(11,20,23,34)(13,29,24,35)(14,18,30,27)(16,28,25,38)(17,31,26,39)(19,22,37,32)(21,36,40,33)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416340);