#512252

order	512
nilpotency class	4
isoclinism	isoclinic class of #511747
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 ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

509 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 509 conjugacy classes of subgroups — too many to draw. Only its 140 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,24,29)(19,25,23,40)(20,26,21,34)(22,36,39,48)(27,33,45,35)(28,31,46,38)(30,41,32,43)(37,42,47,44);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(17,19,24,23)(18,25,29,40)(20,32,21,30)(22,31,39,38)(26,43,34,41)(27,42,45,44)(28,36,46,48)(33,47,35,37);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(17,20,24,21)(18,26,29,34)(19,30,23,32)(22,33,39,35)(25,41,40,43)(27,36,45,48)(28,44,46,42)(31,37,38,47);(17,21,24,20)(18,27,29,45)(19,30,23,32)(22,37,39,47)(25,42,40,44)(26,36,34,48)(28,43,46,41)(31,33,38,35)

Polycyclic presentation:

2^9|a^2=i,b^2=efi,c^2=ei,d^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,g^a=gh,h^a=hi,c^b=ci,g^d=gh,h^d=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, 419448);