#6351848

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	#62823	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2579 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)(3,13)(4,18)(5,11)(6,10)(7,12)(8,16)(9,20)(14,22)(15,24)(17,23)(19,26)(21,25)(27,29)(28,31)(30,32)(33,34,39,43)(35,44,48,58)(36,41,49,55)(37,42,38,51)(40,46,54,47)(45,60,61,59)(50,56,52,57)(53,63,64,62);(1,3,7,17)(2,8,12,24)(4,14,21,30)(5,15,6,16)(9,22,26,32)(10,23,11,13)(18,31,25,27)(19,28,20,29)(33,35,39,48)(34,40,43,54)(36,45,49,61)(37,47,38,46)(41,53,55,64)(42,58,51,44)(50,63,52,62)(56,60,57,59);(1,4)(2,9)(3,14)(5,20)(6,19)(7,21)(8,22)(10,18)(11,25)(12,26)(13,27)(15,29)(16,28)(17,30)(23,31)(24,32)(33,36)(34,41)(35,45)(37,50)(38,52)(39,49)(40,53)(42,56)(43,55)(44,59)(46,62)(47,63)(48,61)(51,57)(54,64)(58,60);(1,5,7,6)(2,10,12,11)(3,15,17,16)(4,19,21,20)(8,23,24,13)(9,25,26,18)(14,28,30,29)(22,27,32,31)(36,49)(41,55)(45,61)(50,52)(53,64)(56,57)(59,60)(62,63);(1,5,7,6)(2,11,12,10)(3,16,17,15)(4,20,21,19)(8,23,24,13)(9,25,26,18)(14,28,30,29)(22,31,32,27)(33,37,39,38)(34,42,43,51)(35,46,48,47)(36,50,49,52)(40,44,54,58)(41,56,55,57)(45,62,61,63)(53,59,64,60)

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=hi,e^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, 6259044);