#151237

order	512
nilpotency class	5
isoclinism	isoclinic class of #151236
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#12137
commutator, Frattini, cosocle	#408	C₂³
prime radical	#3199	C₂²
LCS	G ▷ #408 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #408 ◁ G
LECS	G ▷ #408 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

239 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 239 conjugacy classes of subgroups — too many to draw. Only its 36 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=fh,d^2=fi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=df,f^a=fi,g^a=gh,c^b=ch,d^b=df,e^b=eghi,f^b=fi,g^b=gi,d^c=dg,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=eh,g^e=gi

Conjugacy classes & character table

29 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58433);