#499698

order	512
nilpotency class	4
isoclinism	isoclinic class of #13304
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle, prime radical	C₂³	C₂²×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

500 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=e,d^2=efi,g^2=hi,h^2=i,b^a=be,c^a=cf,c^b=cg,g^b=gh,h^b=hi,g^c=gh,h^c=hi

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, 406894);