#9668297

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#2825	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3739 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gi,d^2=hi,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,e^d=egi,f^d=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9575493);