#145377

order	512
nilpotency class	4
isoclinism	isoclinic class of #143954
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1687
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#10473
prime radical	#33594	C₂
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

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

normal subgroups only

This group has 303 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)(3,14)(4,20)(5,26)(6,12)(7,31)(8,34)(9,16)(10,23)(11,30)(13,29)(15,39)(17,51)(18,38)(19,47)(21,55)(22,40)(24,43)(25,42)(27,46)(28,41)(32,37)(33,45)(35,44)(36,50)(48,64)(49,61)(52,56)(53,59)(54,58)(57,62)(60,63);(1,3)(2,9)(4,21)(5,27)(6,19)(7,32)(8,35)(10,39)(11,44)(12,38)(13,46)(14,26)(15,22)(16,30)(17,29)(18,33)(20,52)(23,57)(24,60)(25,50)(28,49)(31,47)(34,37)(36,41)(40,55)(42,64)(43,61)(45,51)(48,58)(53,63)(54,62)(56,59);(1,4,5,28,29,23,33,59,6,25,7,22,8,24,30,58)(2,10,11,41,34,20,31,54,12,43,45,40,13,42,26,53)(3,15,18,21,35,56,27,60,19,49,16,50,17,48,32,57)(9,36,47,61,46,63,44,52,38,55,14,39,37,62,51,64)

Polycyclic presentation:

2^9|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=fgi,g^a=gi,c^b=cf,d^b=dg,e^b=eg,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, 52573);