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×#58552 |
| 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 | |
4566 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);(4,7)(5,9)(12,16)(13,19)(18,23)(21,25)(27,28,31,32)(29,33,35,39)(30,34,37,40)(36,41,38,42);(3,4,6,7)(5,10,9,8)(14,22)(18,23)(21,25)(24,26)(28,32)(29,35)(30,37)(41,42);(3,4,6,7)(5,8,9,10)(11,12,15,16)(13,20,19,17)(14,18,22,23)(21,26,25,24)(27,29)(28,33)(30,38)(31,35)(32,39)(34,42)(36,37)(40,41);(3,4)(5,8)(6,7)(9,10)(11,13,15,19)(12,17,16,20)(14,21,22,25)(18,24,23,26)(27,30)(28,34)(29,36)(31,37)(32,40)(33,41)(35,38)(39,42);(3,5,6,9)(4,8,7,10)(11,14,15,22)(12,18,16,23)(13,21,19,25)(17,24,20,26)(27,28)(29,33)(30,34)(31,32)(35,39)(36,41)(37,40)(38,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8011522);