Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C27×C4 |
| direct product | C4×#2985 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | C24 |
| commutator | C22 | C23×C42 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C24×C4 |
| prime radical | C24 | C23×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2060 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,3,4);(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,22,24,27,25,28,31,34)(23,26,30,33,29,32,35,36);(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(21,23)(22,26)(24,30)(25,29)(27,33)(28,32)(31,35)(34,36);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19);(6,12)(10,13)(11,20)(18,19)(23,29)(26,32)(30,35)(33,36)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6277825);