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×#27674 |
| 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)(13,21)(14,28)(15,32)(16,23)(17,22)(18,24)(19,26)(20,30)(25,39)(27,35)(29,42)(31,38)(33,37)(34,41)(36,40);(3,5)(4,7)(6,9)(8,10)(11,13,16,26,18,27,17,15)(12,19,22,21,24,32,23,35)(14,25,30,40,31,41,33,29)(20,34,28,36,37,39,38,42);(5,9)(7,10)(11,14)(12,20)(13,25)(15,29)(16,30)(17,33)(18,31)(19,34)(21,36)(22,28)(23,38)(24,37)(26,40)(27,41)(32,39)(35,42);(11,15,17,27,18,26,16,13)(12,21,23,19,24,35,22,32)(14,29,33,41,31,40,30,25)(20,36,38,34,37,42,28,39)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6364434);