#508887

order	512
nilpotency class	4
isoclinism	isoclinic class of #508304
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

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

520 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 158 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,24,41,38,30,39,31,23,27,21,28,22,19,25,42)(20,32,44,46,40,33,37,26,36,48,34,29,35,47,43,45);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,19,21,31,38,18,25,28,23,30,24,42,22,27,39,41)(20,33,34,45,40,48,43,46,36,47,44,26,35,32,37,29);(3,7)(6,8)(10,14)(13,16)(17,20,23,36)(18,26,27,45)(19,29,30,46)(21,37,24,43)(22,40,38,35)(25,44,39,34)(28,47,41,33)(31,32,42,48);(1,3,4,7)(2,6,5,8)(18,27)(19,30)(26,45)(28,41)(29,46)(31,42)(32,48)(33,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=f,c^2=i,d^2=e,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^b=di,d^c=de,f^c=fh,g^c=gh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416083);