#6351812

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351787
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	#62761	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2575 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)(3,13,17,26)(4,18,21,25)(5,10,6,12)(8,15,23,16)(9,19,24,20)(14,22,30,32)(27,28,31,29)(33,34,39,44)(35,45,48,58)(36,41,51,57)(37,43,38,42)(40,52,54,47)(46,60,63,59)(49,56,50,55)(53,62,64,61);(1,3,7,17)(2,8,11,23)(4,14,21,30)(5,15,6,16)(9,22,24,32)(10,13,12,26)(18,31,25,27)(19,28,20,29)(33,35,39,48)(34,40,44,54)(36,46,51,63)(37,47,38,52)(41,53,57,64)(42,45,43,58)(49,61,50,62)(55,60,56,59);(1,4)(2,9)(3,14)(5,20)(6,19)(7,21)(8,22)(10,25)(11,24)(12,18)(13,27)(15,29)(16,28)(17,30)(23,32)(26,31)(33,36)(34,41)(35,46)(37,50)(38,49)(39,51)(40,53)(42,56)(43,55)(44,57)(45,59)(47,62)(48,63)(52,61)(54,64)(58,60);(1,5,7,6)(2,10,11,12)(3,15,17,16)(4,19,21,20)(8,13,23,26)(9,18,24,25)(14,28,30,29)(22,31,32,27)(33,37,39,38)(34,42,44,43)(35,47,48,52)(36,49,51,50)(40,45,54,58)(41,55,57,56)(46,61,63,62)(53,60,64,59);(2,11)(8,23)(9,24)(10,12)(13,26)(18,25)(22,32)(27,31)(33,37,39,38)(34,42,44,43)(35,47,48,52)(36,50,51,49)(40,45,54,58)(41,56,57,55)(46,62,63,61)(53,59,64,60)

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=h,e^2=i,f^2=h,g^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,g^a=ghi,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, 6259008);