Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
| direct product | C22×#1521 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C2×C22⋊C4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
3518 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2);(3,4);(6,11,13,25)(7,16)(8,21)(12,27,28,33)(14,26,15,22)(17,32,29,24)(19,35)(20,31)(23,34)(30,36);(5,6)(7,17)(8,22)(9,14)(10,15)(11,23)(12,20)(13,18)(16,28)(19,32)(21,25)(24,35)(26,34)(27,36)(29,31)(30,33);(5,7,18,31)(6,12,13,28)(8,19,34,35)(9,20,10,16)(11,24,25,32)(14,17,15,29)(21,36,23,30)(22,27,26,33)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250533);