Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#13011 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C23 | C4×D8 |
| prime radical | C22×#508 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1324 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)(7,8,13,20,14,21,11,18,15,22,31,37,23,32,12,19)(9,17,25,16,26,34,10,28,27,33,30,35,29,38,24,36);(3,4,5,6)(7,9,23,26,15,27,14,29)(8,16,32,28,22,35,21,36)(10,12,30,31,24,11,25,13)(17,18,34,20,33,19,38,37);(7,10)(8,17)(9,12)(11,27)(13,29)(14,25)(15,24)(16,18)(19,35)(20,28)(21,38)(22,33)(23,30)(26,31)(32,34)(36,37)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401685);