Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23439 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C2×#23532 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4606 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 38 points:
(1,2);(3,4)(5,6)(7,8)(9,19)(10,24)(11,17)(12,16)(13,18)(14,21)(15,28)(20,29)(22,31)(23,30)(25,37)(26,33)(27,32)(34,36)(35,38);(7,9,13,23)(8,14,18,31)(10,25,26,20)(11,22,12,21)(15,29,32,37)(16,30,17,19)(24,34,33,38)(27,35,28,36);(4,6)(7,10,11,27,13,26,12,28)(8,15,16,33,18,32,17,24)(9,20,21,35,23,25,22,36)(14,29,19,34,31,37,30,38);(10,26)(15,32)(20,25)(24,33)(27,28)(29,37)(34,38)(35,36)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339138);