#136509

order	512
nilpotency class	4
isoclinism	isoclinic class of #136509
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8921
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#3147	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

343 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 343 conjugacy classes of subgroups — too many to draw. Only its 88 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)(3,13,17,42)(4,19,21,9)(5,23)(6,28)(7,11)(8,15,33,50)(10,26)(12,30)(14,43)(16,36,18,51)(20,54,55,37)(22,35,29,40)(24,32,49,39)(25,53,57,34)(27,56)(31,44)(38,58)(41,46)(45,62)(47,59)(48,61)(52,64)(60,63);(1,3,6,16,7,17,30,18)(2,8,12,39,11,33,28,32)(4,14,29,46,21,47,22,45)(5,24,26,15,27,49,58,50)(9,31,40,60,19,52,35,61)(10,36,23,13,38,51,56,42)(20,48,57,64,55,63,25,44)(34,41,54,43,53,62,37,59);(1,4)(2,9)(3,14)(5,25)(6,29)(7,21)(8,31)(10,37)(11,19)(12,40)(13,41)(15,48)(16,46)(17,47)(18,45)(20,26)(22,30)(23,34)(24,44)(27,57)(28,35)(32,61)(33,52)(36,59)(38,54)(39,60)(42,62)(43,51)(49,64)(50,63)(53,56)(55,58)

Polycyclic presentation:

2^9|a^2=f,b^2=fg,d^2=g,e^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,d^b=dg,e^b=eg,f^b=fi,g^b=gi,h^b=hi,d^c=dg,e^c=ehi,f^c=fi,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43705);