#135751

order	512
nilpotency class	4
isoclinism	isoclinic class of #13340
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#493
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#8810
prime radical	C₂³	#430
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

215 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

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

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, 42947);