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×#54679 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4604 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 42 points:
(1,2);(3,4)(5,9)(6,7)(8,10)(11,12)(13,19)(14,16)(15,22)(17,20)(18,23)(21,24)(25,26)(27,28,31,32)(29,35,37,33)(30,38,39,34)(36,41,40,42);(3,5,7,10)(4,8,6,9)(12,14)(13,20)(18,22)(21,26)(28,32)(30,39)(33,35)(36,40);(5,10)(8,9)(11,13)(12,17)(14,19)(15,21)(16,20)(18,24)(22,25)(23,26)(27,29)(28,33)(30,40)(31,37)(32,35)(34,42)(36,39)(38,41);(3,5)(4,9)(6,8)(7,10)(11,14,16,12)(13,19,20,17)(15,22,23,18)(21,25,26,24)(27,30,31,39)(28,34,32,38)(29,36,37,40)(33,41,35,42);(3,6,7,4)(5,8,10,9)(11,15)(12,18)(13,21)(14,22)(16,23)(17,24)(19,25)(20,26)(27,28)(29,33)(30,34)(31,32)(35,37)(36,41)(38,39)(40,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8010887);