#93890

order	512
nilpotency class	4
isoclinism	isoclinic class of #1347
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#102
commutator	C₂×C₈	C₂×C₁₆
Frattini, cosocle	C₂×C₄×C₁₆	C₂²
socle	C₂²	D₄⋊C₁₆
prime radical	C₂×#135	C₈
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₁₆ ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

159 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 159 conjugacy classes of subgroups — too many to draw. Only its 52 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) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 2 generators on 72 points:

(1,2)(3,7,8,5)(4,6)(9,10,13,19,15,21,36,51,14,20,35,50,33,48,62,70,16,22,37,52,38,53,65,72,31,46,61,69,12,18,32,47)(11,23,25,54,27,41,58,49,28,55,59,71,30,17,39,40,29,42,60,66,24,43,56,67,26,45,34,63,57,44,64,68);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,30)(13,34)(14,28)(15,24)(16,29)(18,45)(19,49)(20,43)(21,23)(22,44)(25,35)(26,31)(27,38)(32,58)(33,57)(36,39)(37,59)(40,50)(41,46)(42,53)(47,66)(48,55)(51,63)(52,67)(54,70)(56,62)(60,61)(64,65)(68,69)(71,72)

Polycyclic presentation:

2^9|a^2=d,c^2=fi,d^2=g,f^2=i,g^2=h,h^2=i,b^a=bc,c^a=ce,c^b=cf,d^b=def,f^b=fi,g^b=gi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1086);