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×#57060 |
| 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);(3,4)(5,9)(6,7)(8,10)(11,12,16,13)(14,17,21,19)(15,20,23,18)(22,25,26,24)(27,28)(29,34)(30,31)(32,35)(33,36)(37,41)(38,39)(40,42);(4,7)(8,9)(12,13)(14,21)(15,23)(24,25)(27,29,33,31)(28,34,36,30)(32,39,40,37)(35,38,42,41);(11,13,16,12)(14,17,21,19)(15,20,23,18)(22,24,26,25)(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)(11,15,16,23)(12,18,13,20)(14,22,21,26)(17,24,19,25)(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, 8006882);