#6458778

order	512
nilpotency class	3
isoclinism	isoclinic class of #6442596
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#535
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	C₂×#2482
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2167 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=g,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,g^b=gi,h^b=hi,f^c=fi,e^d=ei

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, 6365974);