#6351939

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
prime radical	#63057	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2483 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)(33,34,39,43)(35,44,47,55)(36,48,50,41)(37,42,38,52)(40,53,56,46)(45,60,63,59)(49,58,51,57)(54,62,64,61);(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)(33,35,39,47)(34,40,43,56)(36,45,50,63)(37,46,38,53)(41,54,48,64)(42,55,52,44)(49,61,51,62)(57,59,58,60);(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)(33,34)(35,40)(36,41)(37,42)(38,52)(39,43)(44,53)(45,54)(46,55)(47,56)(48,50)(49,57)(51,58)(59,61)(60,62)(63,64);(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)(33,36,39,50)(34,41,43,48)(35,45,47,63)(37,51,38,49)(40,54,56,64)(42,58,52,57)(44,59,55,60)(46,62,53,61);(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)(33,37,39,38)(34,42,43,52)(35,46,47,53)(36,49,50,51)(40,55,56,44)(41,57,48,58)(45,61,63,62)(54,59,64,60)

Polycyclic presentation:

2^9|a^2=i,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, 6259135);