Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10176 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
| direct product | C2×#10185 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1560 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2);(3,4)(5,9)(6,8)(7,10)(11,12,17,22)(13,18,25,34)(14,26,28,37)(15,21,16,20)(19,30,36,29)(23,35,39,42)(24,38,31,33)(27,32,40,41);(4,8)(5,6)(9,10)(11,13)(12,18)(14,27)(15,24)(16,31)(17,25)(19,35)(20,33)(21,38)(22,34)(23,30)(26,41)(28,40)(29,39)(32,37)(36,42);(3,5,7,6)(4,9,10,8)(11,14)(12,19)(13,23)(15,30)(16,29)(17,28)(18,32)(20,37)(21,26)(22,36)(24,40)(25,39)(27,31)(33,42)(34,41)(35,38)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 144396);