#6351805

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351787
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	C₄²	C₂⁵
prime radical	C₂×#3553	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2586 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6259001);