#6432812

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432811
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂²×#401
commutator, Frattini, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

1543 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)(3,13,17,8)(4,9,21,32)(5,10)(6,11)(7,12)(14,29)(15,37,42,27)(16,38,44,28)(18,26)(19,30,48,58)(20,31,50,60)(22,33)(23,34)(24,35)(25,36)(39,55)(40,56)(41,54,43,53)(45,51)(46,52)(47,57,49,59)(61,64)(62,63);(1,3)(2,8)(4,18,21,14)(5,15)(6,16)(7,17)(9,29,32,26)(10,27)(11,28)(12,13)(19,45,48,39)(20,46,50,40)(22,41)(23,42)(24,43)(25,44)(30,55,58,51)(31,56,60,52)(33,53)(34,37)(35,54)(36,38)(47,62,49,61)(57,64,59,63);(1,4)(2,9,12,32)(3,14)(5,19)(6,20)(7,21)(8,26,13,29)(10,30,34,58)(11,31,36,60)(15,39)(16,40)(17,18)(22,47)(23,48)(24,49)(25,50)(27,51,37,55)(28,52,38,56)(33,57,35,59)(41,61)(42,45)(43,62)(44,46)(53,63,54,64);(1,5,7,23)(2,10,12,34)(3,15,17,42)(4,19,21,48)(6,24,25,22)(8,27,13,37)(9,30,32,58)(11,35,36,33)(14,39,18,45)(16,43,44,41)(20,49,50,47)(26,51,29,55)(28,54,38,53)(31,59,60,57)(40,62,46,61)(52,64,56,63);(1,6,7,25)(2,11,12,36)(3,16,17,44)(4,20,21,50)(5,22,23,24)(8,28,13,38)(9,31,32,60)(10,33,34,35)(14,40,18,46)(15,41,42,43)(19,47,48,49)(26,52,29,56)(27,53,37,54)(30,57,58,59)(39,61,45,62)(51,63,55,64)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6340008);