#508981

order	512
nilpotency class	4
isoclinism	isoclinic class of #508304
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

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

460 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)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13)(17,18,20,22)(19,24,26,23)(21,25,28,30)(27,29,32,31);(2,5)(6,8)(17,19,21,27,18,23,25,31,20,26,28,32,22,24,30,29);(3,7)(6,8)(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,26)(21,29)(22,23)(25,32)(27,30)(28,31);(1,3,4,7)(2,6,5,8)(19,26)(23,24)(27,32)(29,31)

Polycyclic presentation:

2^9|a^2=fi,b^2=g,c^2=f,d^2=ei,g^2=hi,h^2=i,b^a=be,c^a=cf,c^b=cg,d^b=di,d^c=de,g^c=gh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416177);