#145977

order	512
nilpotency class	4
isoclinism	isoclinic class of #145977
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10601
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

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

normal subgroups only

This group has 198 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:

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

Polycyclic presentation:

2^9|a^2=deh,b^2=dg,c^2=h,d^2=gi,e^2=hi,f^2=gi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,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, 53173);