#6350975

order	512
nilpotency class	3
isoclinism	isoclinic class of #6350909
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)(3,12)(4,9)(5,11)(6,19)(7,10)(8,14)(13,26)(15,23)(16,25)(17,24)(18,29)(20,22)(21,28)(27,31)(30,32)(33,34)(35,46)(36,51)(37,43)(38,44)(39,45)(40,49)(41,53)(42,50)(47,62)(48,57)(52,56)(54,60)(55,59)(58,64)(61,63);(1,3,7,15)(2,8,10,22)(4,13,16,27)(5,14,6,20)(9,21,25,32)(11,23,19,12)(17,28,18,30)(24,31,29,26)(33,35)(34,40)(36,47)(37,48)(38,49)(39,50)(41,56)(42,44)(43,46)(45,57)(51,61)(52,55)(53,63)(54,64)(58,59)(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)(35,47)(37,53)(38,54)(39,55)(40,56)(42,61)(43,59)(44,51)(45,60)(46,58)(48,63)(49,64)(50,52)(57,62);(2,10)(4,16)(8,22)(11,19)(12,23)(13,27)(17,18)(28,30)(33,35)(34,42)(36,52)(37,49)(38,48)(39,50)(40,44)(41,58)(43,57)(45,46)(47,55)(51,62)(53,63)(54,64)(56,59)(60,61);(1,5,7,6)(2,11,10,19)(3,14,15,20)(4,17,16,18)(8,23,22,12)(9,24,25,29)(13,28,27,30)(21,31,32,26)(33,37,39,38)(34,43,45,44)(35,48,50,49)(36,53,55,54)(40,46,57,42)(41,59,60,51)(47,63,52,64)(56,58,62,61)

Polycyclic presentation:

2^9|b^2=hi,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dh,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, 6258171);