Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29602 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
| direct product | C2×#30102 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1506 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 42 points:
(1,2);(3,4)(5,8)(6,7)(9,10)(11,12,15,18)(13,19,21,16)(14,17,22,24)(20,25,26,23)(27,28,31,35)(29,36,39,33)(30,40,32,34)(37,42,38,41);(11,13,15,21)(12,16,18,19)(14,20,22,26)(17,23,24,25)(27,29,31,39)(28,33,35,36)(30,38,32,37)(34,41,40,42);(4,7)(8,10)(11,14,15,22)(12,17,18,24)(13,20,21,26)(16,23,19,25)(29,37)(30,32)(33,41)(34,40)(36,42)(38,39);(3,5,6,9)(4,8,7,10)(14,22)(17,24)(20,26)(23,25)(27,30,31,32)(28,34,35,40)(29,38,39,37)(33,41,36,42)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6341986);