#126286

order	512
nilpotency class	4
isoclinism	isoclinic class of #125887
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1659
commutator, Frattini, cosocle	#376	C₂³
socle	C₂	#7786
prime radical	#31077	C₂
LCS	G ▷ #376 ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #376 ◁ G
LECS	G ▷ #376 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

371 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 371 conjugacy classes of subgroups — too many to draw. Only its 54 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 33482);