#6355034

order	512
nilpotency class	3
isoclinism	isoclinic class of #6354981
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1516 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,d^2=h,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=di,e^b=eh,e^c=eh,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, 6262230);