#6350784

order	512
nilpotency class	3
isoclinism	isoclinic class of #6350717
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
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6257980);