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×#54794 |
| 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);(3,4,9,10)(5,6,7,8)(11,12)(13,16)(14,19)(15,20)(17,21)(18,23)(22,24)(25,26)(27,28,31,32)(29,35,37,33)(30,38,39,34)(36,40,42,41);(3,5,9,7)(4,6,10,8)(12,13)(14,21)(18,20)(22,26)(28,32)(30,39)(33,35)(36,42);(3,6,9,8)(4,5,10,7)(11,13,16,12)(14,19,21,17)(15,20,23,18)(22,25,26,24)(27,29,31,37)(28,33,32,35)(30,36,39,42)(34,40,38,41);(3,7,9,5)(4,6,10,8)(11,14)(12,17)(13,19)(15,22)(16,21)(18,24)(20,25)(23,26)(27,28)(29,35)(30,34)(31,32)(33,37)(36,41)(38,39)(40,42);(3,8,9,6)(4,5,10,7)(11,15,16,23)(12,18,13,20)(14,22,21,26)(17,24,19,25)(27,30,31,39)(28,34,32,38)(29,36,37,42)(33,40,35,41)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8007003);