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×#55850 |
| 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 | |
4588 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);(3,4,7,5)(6,9,10,8)(12,16)(13,19)(18,23)(21,25)(27,28)(29,32)(30,38)(31,34)(33,39)(35,37)(36,41)(40,42);(4,5)(8,9)(14,22)(18,23)(21,25)(24,26)(27,28,31,34)(29,35,37,32)(30,38,39,33)(36,40,42,41);(11,12,15,16)(13,20,19,17)(14,18,22,23)(21,26,25,24)(27,28)(29,35)(30,33)(31,34)(32,37)(36,41)(38,39)(40,42);(3,5,7,4)(6,9,10,8)(11,13,15,19)(12,17,16,20)(14,21,22,25)(18,24,23,26)(27,29,31,37)(28,32,34,35)(30,36,39,42)(33,40,38,41);(3,6,7,10)(4,8,5,9)(11,14)(12,18)(13,21)(15,22)(16,23)(17,24)(19,25)(20,26)(27,30,31,39)(28,33,34,38)(29,36,37,42)(32,40,35,41)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8008895);