Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24278 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#27673 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2028 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 42 points:
(1,2);(3,4,6,8)(5,7,9,10)(11,12)(13,25)(14,29)(15,19)(16,23)(17,22)(18,24)(20,31)(21,28)(26,40)(27,35)(30,39)(32,38)(33,37)(34,41)(36,42);(3,5)(4,7)(6,9)(8,10)(11,13,17,27,18,28,16,15)(12,19,23,21,24,35,22,25)(14,26,33,42,32,41,31,30)(20,34,38,36,37,40,29,39);(5,9)(7,10)(11,14)(12,20)(13,26)(15,30)(16,31)(17,33)(18,32)(19,34)(21,36)(22,29)(23,38)(24,37)(25,39)(27,42)(28,41)(35,40);(11,15,16,28,18,27,17,13)(12,21,22,19,24,25,23,35)(14,30,31,41,32,42,33,26)(20,36,29,34,37,39,38,40)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6364435);