Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#55701 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5054 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(5,9)(7,10)(11,12,14,16)(13,17,18,15)(19,20)(21,27)(22,24)(23,30)(25,28)(26,31)(29,32)(33,34);(3,4,6,8)(5,7,9,10)(13,18)(15,17)(20,22)(21,28)(26,30)(29,34);(3,5,6,9)(4,7,8,10)(11,13,14,18)(12,15,16,17)(19,21)(20,25)(22,27)(23,29)(24,28)(26,32)(30,33)(31,34);(3,5)(4,7)(6,9)(8,10)(11,12)(13,15)(14,16)(17,18)(19,22,24,20)(21,27,28,25)(23,30,31,26)(29,33,34,32);(3,4)(5,7)(6,8)(9,10)(19,23,24,31)(20,26,22,30)(21,29,28,34)(25,32,27,33)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8006392);