#136391

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

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

normal subgroups only

This group has 323 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fh,d^2=gi,e^2=hi,f^2=i,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, 43587);