#6351263

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351101
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	#62943	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2568 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 64 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6258459);