#6349422

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349370
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	#60972	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2573 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)(3,13)(4,9)(5,11)(6,10)(7,12)(8,15)(14,27)(16,22)(17,26)(18,25)(19,23)(20,24)(21,29)(28,32)(30,31)(33,34)(35,40)(36,47)(37,43)(38,50)(39,42)(41,48)(44,59)(45,55)(46,54)(49,58)(51,56)(52,57)(53,62)(60,64)(61,63);(1,3)(2,8)(4,14)(5,15)(6,16)(7,17)(9,21)(10,13)(11,26)(12,22)(18,28)(19,29)(20,30)(23,31)(24,32)(25,27)(33,35)(34,40)(36,44)(37,45)(38,51)(39,46)(41,53)(42,54)(43,55)(47,63)(48,60)(49,61)(50,56)(52,62)(57,64)(58,59);(1,4,7,20)(2,9,12,24)(3,14,17,30)(5,19,6,18)(8,21,22,32)(10,25,11,23)(13,27,26,31)(15,29,16,28)(33,36)(34,41)(35,44)(37,48)(38,52)(39,49)(40,53)(42,57)(43,58)(45,60)(46,61)(47,50)(51,62)(54,64)(55,59)(56,63);(1,5,7,6)(2,10,12,11)(3,15,17,16)(4,18,20,19)(8,13,22,26)(9,23,24,25)(14,28,30,29)(21,31,32,27)(34,42)(40,54)(41,57)(43,50)(47,58)(53,64)(55,56)(59,63);(1,5,7,6)(2,11,12,10)(3,16,17,15)(4,19,20,18)(8,13,22,26)(9,23,24,25)(14,28,30,29)(21,27,32,31)(33,37,39,38)(34,43,42,50)(35,45,46,51)(36,48,49,52)(40,55,54,56)(41,58,57,47)(44,60,61,62)(53,59,64,63)

Polycyclic presentation:

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

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, 6256618);