#136390

order	512
nilpotency class	4
isoclinism	isoclinic class of #136390
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8890
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#135	D₄
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

318 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 94 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)(3,8,14,32)(4,17,20,52)(5,22)(6,10)(7,28)(9,19,35,54)(11,37)(12,25)(13,43)(15,48,47,38)(16,41,29,33)(18,39,21,55)(23,56,57,31)(24,34,53,42)(26,40)(27,36)(30,58)(44,62)(45,61)(46,63)(49,60)(50,51)(59,64);(1,3,7,29,6,14,25,16)(2,8,12,41,10,32,28,33)(4,13,18,44,20,46,21,50)(5,23,27,15,26,57,37,47)(9,30,42,60,35,61,34,59)(11,38,40,31,36,48,22,56)(17,51,55,63,52,62,39,43)(19,49,53,58,54,64,24,45);(1,4)(2,9)(3,13)(5,24)(6,20)(7,18)(8,30)(10,35)(11,39)(12,42)(14,46)(15,49)(16,50)(17,40)(19,27)(21,25)(22,52)(23,45)(26,53)(28,34)(29,44)(31,51)(32,61)(33,59)(36,55)(37,54)(38,43)(41,60)(47,64)(48,63)(56,62)(57,58)

Polycyclic presentation:

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

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, 43586);