#508878

order	512
nilpotency class	4
isoclinism	isoclinic class of #508292
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

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

518 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 170 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,27,37,47,40,48,24,30,22,28,23,29,38,41)(19,25,31,42,35,46,36,26,20,34,32,43,33,44,39,45);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,19,21,31,37,35,40,36,24,20,22,32,23,33,38,39)(18,25,27,42,47,46,48,26,30,34,28,43,29,44,41,45);(3,7)(6,8)(10,14)(13,16)(17,20,24,19)(18,26,30,45)(21,36,22,39)(23,33,37,35)(25,41,34,48)(27,46,28,44)(29,43,47,42)(31,38,32,40);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|a^2=gi,b^2=f,c^2=i,d^2=e,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,c^b=cg,d^c=de,f^c=fh,g^c=gh,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416074);