#145151

order	512
nilpotency class	4
isoclinism	isoclinic class of #145151
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10431
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#28245	C₂
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

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

normal subgroups only

This group has 329 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)(3,10)(4,20)(5,27)(6,31)(7,33)(8,13)(9,14)(11,23)(12,29)(15,47)(16,37)(17,52)(18,39)(19,40)(21,35)(22,43)(24,46)(25,45)(26,56)(28,44)(30,59)(32,60)(34,38)(36,51)(41,49)(42,58)(48,63)(50,61)(53,62)(54,64)(55,57);(1,3)(2,10)(4,21)(5,16)(6,32)(7,18)(8,34)(9,19)(11,41)(12,36)(13,38)(14,40)(15,24)(17,30)(20,55)(22,57)(23,54)(25,53)(26,50)(27,37)(28,48)(29,51)(31,60)(33,39)(35,43)(42,64)(44,47)(45,61)(46,63)(49,58)(52,59)(56,62);(1,4,7,26,9,22,8,25)(2,11,13,46,14,42,33,44)(3,15,18,54,19,48,34,49)(5,28,29,58,30,24,6,23)(10,35,38,62,40,55,39,61)(12,45,27,43,31,56,59,20)(16,50,51,57,17,53,32,21)(36,41,37,63,60,64,52,47)

Polycyclic presentation:

2^9|c^2=dg,e^2=gi,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fi,g^a=gi,c^b=cf,f^b=fh,h^b=hi,d^c=di,e^c=egi,f^c=fh,g^c=gi,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, 52347);