#6351831

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351817
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	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2546 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,d^2=h,f^2=h,g^2=hi,b^a=bf,c^a=cg,d^a=dhi,f^a=fh,g^a=ghi,e^b=eh,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, 6259027);