#512509

order	512
nilpotency class	4
isoclinism	isoclinic class of #511948
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

530 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 234 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)(19,31)(20,25)(21,39)(22,42)(23,43)(24,30)(26,35)(27,32)(28,41)(29,40)(33,45)(34,44)(36,48)(37,47)(38,46);(2,5)(6,8)(17,19,32,37,23,34,41,38,24,35,22,33,40,36,21,20)(18,25,39,48,29,45,42,26,30,46,28,44,43,47,27,31);(9,11,12,16)(10,13,14,15);(1,3,4,7)(2,6,5,8)(17,20,21,36,40,33,22,35,24,38,41,34,23,37,32,19)(18,26,27,45,43,48,28,25,30,31,42,47,29,44,39,46)

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=f,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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419705);