#137749

order	512
nilpotency class	4
isoclinism	isoclinic class of #13096
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1898
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂	#9081
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#568 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

508 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 102 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dgi,e^b=ei,f^b=fg,g^b=gi,d^c=di,e^c=ei,f^c=fi,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44945);