#136511

order	512
nilpotency class	4
isoclinism	isoclinic class of #136511
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)(3,13,17,43)(4,18,20,9)(5,22)(6,27)(7,11)(8,15,33,49)(10,25)(12,30)(14,44)(16,42,28,37)(19,38,55,64)(21,36,29,40)(23,34,59,32)(24,35,56,52)(26,57)(31,45)(39,54)(41,50)(46,63)(47,60)(48,62)(51,58)(53,61);(1,3,6,28,7,17,30,16)(2,8,12,32,11,33,27,34)(4,14,29,46,20,47,21,50)(5,23,25,15,26,59,54,49)(9,31,40,62,18,51,36,61)(10,37,22,13,39,42,57,43)(19,53,24,58,55,48,56,45)(35,63,38,44,52,41,64,60);(1,4)(2,9)(3,14)(5,24)(6,29)(7,20)(8,31)(10,38)(11,18)(12,40)(13,41)(15,48)(16,50)(17,47)(19,54)(21,30)(22,52)(23,58)(25,55)(26,56)(27,36)(28,46)(32,62)(33,51)(34,61)(35,57)(37,44)(39,64)(42,60)(43,63)(45,59)(49,53)

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=egi,f^b=fi,g^b=gi,h^b=hi,d^c=dgi,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, 43707);