#494803

order	512
nilpotency class	4
isoclinism	isoclinic class of #33638
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#476
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1920
prime radical	C₂⁴	#137
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

571 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401999);