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×#58972 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4354 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 6 generators on 42 points:
(1,2);(3,4,6,7)(5,9,10,8)(11,12)(13,16)(14,19)(15,20)(17,21)(18,23)(22,24)(25,26)(27,28)(29,34)(30,31)(32,35)(33,36)(37,41)(38,39)(40,42);(4,7)(5,10)(12,13)(14,21)(18,20)(22,26)(27,29,33,31)(28,34,36,30)(32,39,40,37)(35,38,42,41);(4,7)(8,9)(11,13,16,12)(14,19,21,17)(15,20,23,18)(22,25,26,24)(27,30)(28,29)(31,36)(32,38)(33,34)(35,37)(39,42)(40,41);(3,4)(5,8)(6,7)(9,10)(11,14)(12,17)(13,19)(15,22)(16,21)(18,24)(20,25)(23,26)(27,31,33,29)(28,34,36,30)(32,39,40,37)(35,41,42,38);(3,5,6,10)(4,8,7,9)(11,15)(12,18)(13,20)(14,22)(16,23)(17,24)(19,25)(21,26)(27,32,33,40)(28,35,36,42)(29,37,31,39)(30,38,34,41)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8012467);