#6352807

order	512
nilpotency class	3
isoclinism	isoclinic class of #6352701
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2603 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=i,e^2=i,f^2=h,b^a=bf,c^a=cg,d^a=di,f^a=fh,f^b=fh,g^b=gi,f^c=fi,e^d=eh

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6260003);