#508889

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	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

517 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 517 conjugacy classes of subgroups — too many to draw. Only its 154 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) = 256

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

Polycyclic presentation:

2^9|a^2=gi,b^2=f,d^2=ei,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, 416085);