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×#56874 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4664 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)(6,8)(7,10)(11,12,15,16)(13,19,20,17)(14,22,23,18)(21,24,26,25)(27,28,32,35)(29,31,34,40)(30,33,39,42)(36,38,37,41);(3,4,6,8)(5,7,10,9)(12,16)(13,20)(18,22)(21,26)(29,34)(31,40)(36,37)(38,41);(11,13,15,20)(12,17,16,19)(14,21,23,26)(18,24,22,25)(27,29,32,34)(28,31,35,40)(30,37,39,36)(33,41,42,38);(3,4)(5,7)(6,8)(9,10)(11,12)(13,17)(14,18)(15,16)(19,20)(21,24)(22,23)(25,26)(27,30)(28,33)(29,36)(31,38)(32,39)(34,37)(35,42)(40,41);(3,5,6,10)(4,7,8,9)(11,14)(12,18)(13,21)(15,23)(16,22)(17,24)(19,25)(20,26)(27,31)(28,34)(29,35)(30,38)(32,40)(33,37)(36,42)(39,41)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8006728);