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×#54079 |
| 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 | |
4620 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);(5,9)(7,10)(11,12)(13,19)(14,16)(15,22)(17,20)(18,23)(21,24)(25,26)(27,28,32,33)(29,31,37,34)(30,38,39,35)(36,42,40,41);(3,4,6,8)(5,7,9,10)(12,14)(13,20)(18,22)(21,26)(28,33)(30,39)(31,34)(36,40);(3,5,6,9)(4,7,8,10)(11,13)(12,17)(14,19)(15,21)(16,20)(18,24)(22,25)(23,26)(27,29)(28,34)(30,40)(31,33)(32,37)(35,41)(36,39)(38,42);(3,5)(4,7)(6,9)(8,10)(11,14,16,12)(13,19,20,17)(15,22,23,18)(21,25,26,24)(27,30,32,39)(28,35,33,38)(29,36,37,40)(31,41,34,42);(3,4)(5,7)(6,8)(9,10)(11,15,16,23)(12,18,14,22)(13,21,20,26)(17,24,19,25)(27,31,32,34)(28,29,33,37)(30,41,39,42)(35,36,38,40)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8006341);