#6349735

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349684
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	#62831	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2601 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:

(1,2,6,9)(3,11,13,24)(4,8,15,22)(5,17,19,10)(7,18,21,14)(12,25,26,32)(16,29,28,23)(20,30,31,27)(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);(1,3,6,13)(2,7,9,21)(4,12,15,26)(5,18,19,14)(8,20,22,31)(10,24,17,11)(16,30,28,27)(23,32,29,25)(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)(2,8)(3,12)(5,16)(6,15)(7,20)(9,22)(10,23)(11,25)(13,26)(14,27)(17,29)(18,30)(19,28)(21,31)(24,32)(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);(2,9)(4,15)(7,21)(10,17)(11,24)(12,26)(16,28)(27,30)(34,42)(40,54)(41,57)(43,50)(47,58)(53,64)(55,56)(59,63);(2,9)(3,13)(8,22)(10,17)(12,26)(14,18)(23,29)(27,30)(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|a^2=h,b^2=h,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dhi,e^a=eh,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

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6256931);