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×#54683 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C26 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4830 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 42 points:
(1,2);(3,4)(5,7)(6,9)(8,10)(11,12,16,17)(13,14,21,18)(15,23,24,19)(20,25,26,22)(27,28,31,34)(29,35,37,32)(30,33,38,40)(36,41,42,39);(4,5)(8,9)(12,17)(14,18)(15,24)(20,26)(27,28,31,34)(29,32,37,35)(30,33,38,40)(36,39,42,41);(3,4,7,5)(6,8,10,9)(11,13,16,21)(12,18,17,14)(15,20,24,26)(19,25,23,22)(27,28)(29,32)(30,33)(31,34)(35,37)(36,39)(38,40)(41,42);(3,5,7,4)(6,9,10,8)(11,14,16,18)(12,13,17,21)(15,22,24,25)(19,20,23,26)(27,29,31,37)(28,32,34,35)(30,36,38,42)(33,39,40,41);(3,6)(4,8)(5,9)(7,10)(11,15)(12,19)(13,20)(14,22)(16,24)(17,23)(18,25)(21,26)(27,30,31,38)(28,33,34,40)(29,36,37,42)(32,39,35,41)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8006366);