#134946

order	512
nilpotency class	4
isoclinism	isoclinic class of #134946
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8642
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

285 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 285 conjugacy classes of subgroups — too many to draw. Only its 104 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)(3,15,19,43)(4,21)(5,12)(6,28)(7,31)(8,14)(9,17,34,49)(10,22)(11,26)(13,30)(16,44,46,60)(18,51,50,35)(20,54,53,33)(23,40)(24,56)(25,36)(27,41)(29,37)(32,47,59,48)(38,57)(39,58)(42,62,55,45)(52,61,64,63);(1,3,5,17,7,20,26,50,8,19,27,49,30,53,6,18)(2,9,11,15,13,35,12,33,14,34,28,43,31,51,41,54)(4,16,22,45,23,47,57,64,25,46,58,62,24,48,29,52)(10,32,36,42,37,60,40,61,39,59,21,55,38,44,56,63);(1,4)(2,10)(3,16)(5,22)(6,29)(7,23)(8,25)(9,32)(11,36)(12,40)(13,37)(14,39)(15,42)(17,45)(18,52)(19,46)(20,47)(21,28)(24,30)(26,57)(27,58)(31,38)(33,61)(34,59)(35,60)(41,56)(43,55)(44,51)(48,53)(49,62)(50,64)(54,63)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42142);