#6351926

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

2554 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=hi,d^2=h,e^2=h,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, 6259122);