#511424

order	512
nilpotency class	4
isoclinism	isoclinic class of #509081
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

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 ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

465 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 465 conjugacy classes of subgroups — too many to draw. Only its 132 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 418620);