#145929

order	512
nilpotency class	4
isoclinism	isoclinic class of #145928
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10591
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#471
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

238 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 74 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:

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

Polycyclic presentation:

2^9|a^2=de,b^2=dg,c^2=i,d^2=gi,e^2=h,f^2=gi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,f^a=fgi,g^a=gi,h^a=hi,c^b=cf,f^b=fi,d^c=dg,e^c=eh,f^c=fg,g^c=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53125);