#512515

order	512
nilpotency class	4
isoclinism	isoclinic class of #511954
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 218 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 218 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)(9,10,12,14)(11,15,16,13)(17,18,22,26)(19,27,32,44)(20,25,36,41)(21,29,38,40)(23,28,31,45)(24,39,37,30)(33,46,47,43)(34,48,35,42);(2,5)(6,8)(17,19,31,47,24,34,21,20,22,32,23,33,37,35,38,36)(18,25,40,48,30,43,28,27,26,41,29,42,39,46,45,44);(9,11,12,16)(10,13,14,15)(18,26)(25,41)(27,44)(28,45)(29,40)(30,39)(42,48)(43,46);(1,3,4,7)(2,6,5,8)(17,20,21,35,37,47,31,32,22,36,38,34,24,33,23,19)(18,27,28,46,39,48,40,41,26,44,45,43,30,42,29,25)

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=fi,d^2=e,e^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^a=ghi,h^a=hi,d^b=di

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, 419711);