Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2081 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#11134 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #397 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2618 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 38 points:
(1,2);(7,8,12,16,13,17,11,18)(9,19,23,31,22,30,21,14)(10,24,27,34,28,15,26,33)(20,32,36,29,35,37,25,38);(3,4)(5,6)(7,9,11,21,13,22,12,23)(8,14,16,19,17,31,18,30)(10,25,26,35,28,36,27,20)(15,32,33,29,24,37,34,38);(4,6)(7,10,13,28)(8,15,17,24)(9,20,22,35)(11,27,12,26)(14,29,31,38)(16,34,18,33)(19,32,30,37)(21,36,23,25)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265018);