#145443

order	512
nilpotency class	4
isoclinism	isoclinic class of #144075
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1695
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#10480
prime radical	C₂×#135	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 252 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:

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

Polycyclic presentation:

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

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, 52639);