#145441

order	512
nilpotency class	4
isoclinism	isoclinic class of #145441
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10480
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#135	D₄
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

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

normal subgroups only

This group has 251 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:

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

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=fg,g^a=gi,c^b=cf,d^b=dgi,e^b=eg,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, 52637);