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×#54210 |
| 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 | |
5022 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);(3,4,8,9)(5,7,6,10)(12,15)(16,17)(19,20)(21,27)(22,24)(23,30)(25,29)(26,31)(28,32)(33,34);(4,9)(5,6)(11,12,14,15)(13,17,18,16)(20,22)(23,31)(25,27)(28,34);(3,5,8,6)(4,10,9,7)(11,13,14,18)(12,16,15,17)(19,21,24,29)(20,25,22,27)(23,28,31,34)(26,32,30,33);(3,6,8,5)(4,7,9,10)(11,13)(12,16)(14,18)(15,17)(19,22,24,20)(21,25,29,27)(23,30,31,26)(28,32,34,33);(3,7)(4,5)(6,9)(8,10)(11,12)(13,16)(14,15)(17,18)(19,23)(20,26)(21,28)(22,30)(24,31)(25,32)(27,33)(29,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8010780);