Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32712 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×#32846 |
| 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 | |
1504 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);(3,4,6,8)(5,7,9,10)(11,12)(13,17)(14,18)(15,16)(19,25)(20,23)(21,24)(22,26)(27,28)(29,32)(30,34)(31,33)(35,39)(36,41)(37,40)(38,42);(3,5)(4,7)(6,9)(8,10)(12,16)(14,21)(17,23)(19,22)(27,29,31,37)(28,32,33,40)(30,35,38,36)(34,39,42,41);(4,8)(7,10)(11,13,15,20)(12,17,16,23)(14,22,21,19)(18,26,24,25)(29,35)(30,38)(32,39)(34,42)(36,37)(40,41);(5,9)(7,10)(11,14,15,21)(12,18,16,24)(13,19,20,22)(17,25,23,26)(28,33)(32,40)(34,42)(39,41)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6538937);