#508886

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

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

normal subgroups only

This group has 512 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)(17,18,22,29,41,31,42,32,24,27,21,28,23,19,25,33)(20,34,37,47,44,48,40,26,39,46,36,30,38,35,43,45);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,19,21,32,41,18,25,28,24,31,22,33,23,27,42,29)(20,35,36,26,44,34,43,30,39,48,37,45,38,46,40,47);(3,7)(6,8)(10,14)(13,16)(17,20,24,39)(18,26,27,45)(19,30,31,47)(21,40,22,43)(23,44,41,38)(25,37,42,36)(28,35,29,48)(32,34,33,46);(1,3,4,7)(2,6,5,8)(18,27)(19,31)(26,45)(28,29)(30,47)(32,33)(34,46)(35,48)

Polycyclic presentation:

2^9|a^2=g,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, 416082);