#499673

order	512
nilpotency class	4
isoclinism	isoclinic class of #498795
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

572 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406869);