#126388

order	512
nilpotency class	4
isoclinism	isoclinic class of #126387
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#7795
commutator, Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂×#135	D₄
LCS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#97 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

267 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 267 conjugacy classes of subgroups — too many to draw. Only its 52 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 33584);