#145063

order	512
nilpotency class	4
isoclinism	isoclinic class of #145063
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10413
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

278 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 72 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)(3,13,17,8)(4,19)(5,24)(6,29)(7,10)(9,21)(11,27)(12,31)(14,45,47,61)(15,33,18,35)(16,40,50,63)(20,36,55,53)(22,57)(23,38)(25,34,51,44)(26,54)(28,59)(30,42)(32,46,43,52)(37,60)(39,48)(41,56)(49,64,58,62);(1,3)(2,8)(4,20)(5,25)(6,15)(7,17)(9,36)(10,13)(11,40)(12,33)(14,22)(16,48)(18,31)(19,53)(21,55)(23,58)(24,44)(26,52)(27,50)(28,51)(29,35)(30,49)(32,37)(34,59)(38,64)(39,63)(41,45)(42,62)(43,54)(46,60)(47,56)(57,61);(1,4)(2,9,10,19)(3,14,17,47)(5,26)(6,30)(7,21)(8,32)(11,41,39,57)(12,42,29,38)(13,43)(15,46,18,52)(16,49,50,58)(20,25,55,51)(22,27)(23,31)(24,37,59,54)(28,60)(33,61)(34,62)(35,45)(36,63)(40,53)(44,64)(48,56)

Polycyclic presentation:

2^9|a^2=g,c^2=dg,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,f^a=fh,h^a=hi,c^b=cf,e^b=eh,f^b=fhi,g^b=gi,h^b=hi,e^c=egi,f^c=fh,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, 52259);