#6351855

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351817
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	#62828	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2610 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:

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

Polycyclic presentation:

2^9|a^2=h,e^2=hi,f^2=h,g^2=hi,b^a=bf,c^a=cg,d^a=dhi,f^a=fh,g^a=ghi,e^b=eh,f^b=fh,g^b=gi,d^c=dh,f^c=fi,g^c=ghi

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, 6259051);