#494094

order	512
nilpotency class	4
isoclinism	isoclinic class of #483412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#427
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1835
prime radical	C₂⁴	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

533 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fh,c^2=e,e^2=hi,f^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,h^a=hi,d^b=di,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401290);