#508880

order	512
nilpotency class	4
isoclinism	isoclinic class of #508292
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

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

normal subgroups only

This group has 531 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) = 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)(17,18,22,28,41,19,31,33,21,27,23,29,24,30,25,34)(20,35,38,48,44,36,43,45,37,46,39,32,40,47,42,26);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,19,22,33,41,27,31,29,21,30,23,34,24,18,25,28)(20,36,38,45,44,46,43,32,37,47,39,26,40,35,42,48);(3,7)(6,8)(10,14)(13,16)(17,20)(18,26)(19,32)(21,37)(22,42)(23,43)(24,44)(25,38)(27,45)(28,47)(29,36)(30,48)(31,39)(33,46)(34,35)(40,41);(1,3,4,7)(2,6,5,8)(20,37)(26,45)(32,48)(35,46)(36,47)(38,39)(40,44)(42,43)

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,c^b=cg,d^c=de,f^c=fh,g^c=gh,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416076);