#136873

order	512
nilpotency class	4
isoclinism	isoclinic class of #136871
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8977
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#33594	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

409 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 409 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,d^2=gi,e^2=hi,f^2=gi,h^2=i,b^a=bd,c^a=ce,d^a=di,d^b=dg,e^b=eh,f^b=fg,h^b=hi,d^c=dh,e^c=eh,f^c=fh,g^c=gi,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44069);