#6352028

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351885
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

2531 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:

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

Polycyclic presentation:

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

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, 6259224);