#510366

order	512
nilpotency class	4
isoclinism	isoclinic class of #509065
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 ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

504 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 130 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)(3,5)(4,7)(6,8)(9,10)(11,13)(12,14)(15,16)(18,23)(19,27)(25,39)(26,36)(29,40)(30,37)(33,46)(34,48);(1,3)(2,4)(5,8)(6,7)(9,11,12,15)(10,13,14,16)(17,18,20,23)(19,28,27,45)(21,33,31,46)(22,36,32,26)(24,40,38,29)(25,42,39,35)(30,47,37,43)(34,44,48,41);(1,4,6,5)(2,7,8,3)(10,14)(13,16)(17,19,20,27)(18,24,23,38)(21,34,31,48)(22,37,32,30)(25,43,39,47)(26,44,36,41)(28,46,45,33)(29,42,40,35);(1,5,6,4)(2,7,8,3)(10,14)(13,16)(18,23)(24,38)(25,39)(26,36)(28,45)(33,46)(41,44)(43,47)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 417562);