#143261

order	512
nilpotency class	4
isoclinism	isoclinic class of #143260
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10048
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

396 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 76 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=g,e^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=cf,d^b=dg,e^b=eh,f^b=fi,h^b=hi,d^c=dh,e^c=eh,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, 50457);