#6351925

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

2500 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)(3,13)(4,16)(5,9)(6,10)(7,11)(8,12)(14,27)(15,19)(17,23)(18,28)(20,24)(21,25)(22,26)(29,31)(30,32)(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)(2,4)(5,14)(6,15)(7,16)(8,17)(9,18)(10,13)(11,23)(12,19)(20,28)(21,30)(22,29)(24,31)(25,27)(26,32)(33,35,40,48)(34,41,37,46)(36,45,51,64)(38,47,39,53)(42,55,49,61)(43,56,52,44)(50,62,54,63)(57,60,58,59);(1,4)(2,3)(5,18)(6,13)(7,23)(8,19)(9,14)(10,15)(11,16)(12,17)(20,31)(21,27)(22,32)(24,28)(25,30)(26,29)(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);(1,5,8,22)(2,9,12,26)(3,14,17,29)(4,18,19,32)(6,21,7,20)(10,25,11,24)(13,27,23,31)(15,30,16,28)(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)(2,10,12,11)(3,15,17,16)(4,13,19,23)(5,20,22,21)(9,24,26,25)(14,28,29,30)(18,31,32,27)(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=i,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, 6259121);