Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C42 |
| direct product | C4×#1976 |
| subgroup | G/H | |
|---|---|---|
| center | C42 | #121 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#390 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1600 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)(7,11,16,25)(8,18)(9,21)(10,13)(12,19)(14,22)(15,27,28,36)(17,31,30,24)(20,32)(23,34,35,29)(26,33);(5,7)(6,11,13,25)(8,15)(9,17)(10,16)(12,23,26,35)(14,24,21,31)(18,27,32,36)(19,29)(20,28)(22,30)(33,34);(5,8,9,19,10,20,22,33)(6,12,14,18,13,26,21,32)(7,15,17,29,16,28,30,34)(11,23,24,27,25,35,31,36)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261800);