Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9091 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#11184 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2522 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)(9,19)(10,24,28,33)(11,17)(12,16)(13,18)(14,21)(15,26,34,27)(20,29,37,38)(22,31)(23,30)(25,35,36,32);(3,4)(5,6)(7,9,13,23)(8,14,18,31)(10,25,28,36)(11,22,12,21)(15,32,34,35)(16,30,17,19)(20,26,37,27)(24,29,33,38);(4,6)(7,10,11,26,13,28,12,27)(8,15,16,33,18,34,17,24)(9,20,21,36,23,37,22,25)(14,29,19,35,31,38,30,32)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265134);