#507503

order	512
nilpotency class	4
isoclinism	isoclinic class of #501046
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

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

484 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 190 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=egi,c^2=fh,d^2=ei,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cg,d^c=di

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, 414699);