Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#13018 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #447 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1346 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);(4,6)(7,8,13,20,16,25,12,24,15,21,11,18,14,9,22,19)(10,26,29,17,33,35,28,23,31,38,27,37,30,34,32,36);(3,4,5,6)(7,9,15,25)(8,16,21,14)(10,23,31,36)(11,20,13,18)(12,24,22,19)(17,30,37,33)(26,32,38,28)(27,34,29,35);(7,10,15,31)(8,17,21,37)(9,23,25,36)(11,28,13,32)(12,27,22,29)(14,33,16,30)(18,38,20,26)(19,35,24,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401711);