#153214

order	512
nilpotency class	5
isoclinism	isoclinic class of #153213
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2226
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂×#508	C₂²
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 247 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) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

(5,6,8,13)(7,19)(9,27,25,50)(10,18,12,16)(11,33,26,45)(14,39)(15,43,37,31)(17,29,38,35)(20,49)(21,36)(22,59)(23,52)(24,54)(28,48)(30,62,32,61)(34,53)(40,58)(41,55)(42,64)(44,56)(46,65,47,66)(51,57)(60,68)(63,67);(1,2)(3,4)(6,13)(7,20,24,23)(14,40,36,41)(15,37)(16,18)(17,38)(19,49,54,52)(21,55,39,58)(22,56,60,34)(27,50)(28,63,57,64)(33,45)(42,48,67,51)(44,68,53,59)(46,47)(61,62);(2,4)(5,7)(6,14)(8,24)(9,28)(10,23)(11,34)(12,20)(13,36)(15,44)(16,40)(17,48)(18,41)(19,50)(21,43)(22,35)(25,57)(26,56)(27,54)(29,60)(30,64)(31,39)(32,63)(33,42)(37,53)(38,51)(45,67)(46,68)(47,59)(49,62)(52,61)(55,65)(58,66)

Polycyclic presentation:

2^9|a^2=d,b^2=di,e^2=ghi,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fi,g^a=gh,h^a=hi,c^b=cf,d^c=di,e^c=egi,f^c=fi,g^c=gh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60410);