#6352022

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

2587 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)(33,34,38,42)(35,43,46,56)(36,47,48,40)(37,50,52,41)(39,51,54,45)(44,58,59,57)(49,55,63,62)(53,60,61,64);(1,3)(2,9)(4,14)(5,15)(6,16)(7,17)(8,18)(10,24)(11,13)(12,26)(19,32)(20,28)(21,29)(22,31)(23,30)(25,27)(33,35)(34,39)(36,44)(37,51)(38,46)(40,53)(41,56)(42,54)(43,50)(45,52)(47,61)(48,59)(49,64)(55,58)(57,62)(60,63);(1,4)(2,10)(3,14)(5,20)(6,21)(7,23)(8,22)(9,24)(11,25)(12,19)(13,27)(15,28)(16,29)(17,30)(18,31)(26,32)(37,52)(41,50)(43,56)(45,51)(49,63)(55,62)(57,58)(60,64);(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)(33,36,38,48)(34,40,42,47)(35,44,46,59)(37,49,52,63)(39,53,54,61)(41,55,50,62)(43,57,56,58)(45,60,51,64);(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)(36,48)(40,47)(44,59)(49,63)(53,61)(55,62)(57,58)(60,64)

Polycyclic presentation:

2^9|a^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, 6259218);