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×#55913 |
| 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 | |
5420 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)(5,7)(6,9)(8,10)(12,15)(13,17)(19,20,24,27)(21,22,29,25)(23,31,32,26)(28,33,34,30);(4,5)(6,10)(12,15)(16,18)(19,21,24,29)(20,25,27,22)(23,28,32,34)(26,33,31,30);(4,5)(8,9)(11,12,14,15)(13,16,17,18)(19,21)(20,25)(22,27)(23,28)(24,29)(26,33)(30,31)(32,34);(3,5,7,4)(6,9,10,8)(11,13,14,17)(12,16,15,18)(19,22,24,25)(20,21,27,29)(23,30,32,33)(26,28,31,34);(3,6)(4,8)(5,9)(7,10)(19,23)(20,26)(21,28)(22,30)(24,32)(25,33)(27,31)(29,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8010058);