Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
| direct product | C4×#1881 |
| subgroup | G/H | |
|---|---|---|
| center | C42 | #120 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#386 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1820 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)(7,16,19,27)(8,20)(9,14)(10,24)(11,15)(12,18,28,23)(13,21)(17,32,35,26)(22,29)(25,30)(31,33,36,34);(5,7,9,23,11,19,10,18)(6,12,14,27,15,28,24,16)(8,17,21,34,22,35,25,33)(13,26,29,36,30,32,20,31);(5,8)(6,13)(7,17)(9,21)(10,25)(11,22)(12,26)(14,29)(15,30)(16,31)(18,33)(19,35)(20,24)(23,34)(27,36)(28,32)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261442);