#508961

order	512
nilpotency class	4
isoclinism	isoclinic class of #508292
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

508 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 170 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)(9,10,12,14)(11,15,16,13)(17,18,22,26)(19,23,30,37)(20,24,33,40)(21,25,32,39)(27,34,42,46)(28,35,41,45)(29,36,43,47)(31,38,44,48);(2,5)(6,8)(17,19,20,28,32,43,44,42,22,30,33,41,21,29,31,27)(18,23,24,35,39,47,48,46,26,37,40,45,25,36,38,34);(1,3)(2,6)(4,7)(5,8)(9,11,12,16)(10,13,14,15)(19,27)(20,31)(21,32)(23,34)(24,38)(25,39)(28,29)(30,42)(33,44)(35,36)(37,46)(40,48)(41,43)(45,47);(3,7)(6,8)

Polycyclic presentation:

2^9|a^2=fi,b^2=g,c^2=f,g^2=hi,h^2=i,b^a=be,c^a=cf,c^b=cg,d^c=de,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, 416157);