Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #2714 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3741 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 64 points:
(2,3)(7,9)(8,10)(14,15)(17,18,23,20)(19,24,29,27)(21,25,28,30)(22,26,32,31)(34,39)(37,49)(40,43)(41,50)(45,59)(48,60)(52,55)(58,64);(1,2,6,3)(4,7,11,9)(5,8,13,10)(12,14,16,15)(17,19,23,29)(18,24,20,27)(21,22,28,32)(25,26,30,31)(36,47)(41,50)(44,57)(48,60)(52,55)(54,61)(58,64)(62,63);(4,11)(7,9)(12,16)(14,15)(17,18,23,20)(19,27,29,24)(21,30,28,25)(22,26,32,31)(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);(1,3,6,2)(4,7,11,9)(5,8,13,10)(12,15,16,14)(17,20,23,18)(19,27,29,24)(21,25,28,30)(22,26,32,31)(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);(1,4)(2,7)(3,9)(5,12)(6,11)(8,14)(10,15)(13,16)(17,21,23,28)(18,25,20,30)(19,22,29,32)(24,26,27,31)(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);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)(17,22)(18,26)(19,28)(20,31)(21,29)(23,32)(24,30)(25,27)(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=gi,c^2=i,d^2=ghi,e^2=hi,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=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9575483);