#493927

order	512
nilpotency class	4
isoclinism	isoclinic class of #481288
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#418
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1824
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

971 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=eh,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,f^a=fh,h^a=hi,f^b=fi,g^b=gi,e^c=ei,f^c=fh,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, 401123);