#512473

order	512
nilpotency class	4
isoclinism	isoclinic class of #511948
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

446 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 194 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)(3,6,7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,27)(20,32)(21,24)(22,35)(23,29)(25,28)(26,33)(30,39)(31,46)(34,47)(36,45)(37,48)(38,43)(40,42)(41,44);(2,5)(6,8)(9,11,12,15)(10,13,14,16);(17,19,28,42,22,31,34,45,21,30,43,48,33,44,20,29)(18,23,32,41,26,37,38,39,24,36,47,46,35,40,25,27);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(18,24)(23,36)(25,38)(26,35)(27,39)(32,47)(37,40)(41,46)

Polycyclic presentation:

2^9|a^2=e,b^2=gi,c^2=fh,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,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, 419669);