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×#55758 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5366 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(3,4)(5,9)(6,7)(8,10)(11,12,14,17)(13,16,18,15)(19,20,23,26)(21,27,29,24)(22,25,30,32)(28,33,34,31);(4,7)(8,9)(11,13,14,18)(12,15,17,16)(22,30)(25,32)(28,34)(31,33);(13,18)(15,16)(19,20)(21,27)(22,25)(23,26)(24,29)(28,33)(30,32)(31,34);(3,4)(5,8)(6,7)(9,10)(11,13)(12,16)(14,18)(15,17)(19,21,23,29)(20,24,26,27)(22,28,30,34)(25,31,32,33);(3,5)(4,8)(6,10)(7,9)(11,12)(13,16)(14,17)(15,18)(19,22)(20,25)(21,28)(23,30)(24,31)(26,32)(27,33)(29,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8010031);