#510495

order	512
nilpotency class	4
isoclinism	isoclinic class of #509065
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₄	D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

632 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 632 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 417691);