Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1828 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
| direct product | C4×#1830 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #381 |
| commutator | C23 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#381 |
| prime radical | C24 | C23×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1417 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4);(5,6,9,14,10,13,23,28)(7,15,17,29,18,11,22,25)(8,19,21,33,20,12,26,27)(16,24,32,35,31,30,34,36);(5,7,10,18)(6,11,13,15)(8,16,20,31)(9,22,23,17)(12,24,19,30)(14,29,28,25)(21,34,26,32)(27,36,33,35);(5,8)(6,12,13,19)(7,16)(9,21)(10,20)(11,24,15,30)(14,27,28,33)(17,32)(18,31)(22,34)(23,26)(25,35,29,36)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 262051);