#145381

order	512
nilpotency class	4
isoclinism	isoclinic class of #145380
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10473
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

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

normal subgroups only

This group has 299 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=dh,d^2=gi,e^2=h,f^2=gi,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=dg,e^b=egi,f^b=fg,g^b=gi,h^b=hi,d^c=di,e^c=egh,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, 52577);