Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1909 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
| direct product | C4×#1909 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#386 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1839 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)(7,15)(8,19,21,12)(9,22)(10,14)(11,17)(13,24)(16,29,32,36)(18,30)(20,27,23,28)(25,31,35,33)(26,34);(5,7,9,17,10,18,24,34)(6,11,13,15,14,26,22,30)(8,16,23,33,21,32,20,31)(12,25,28,29,19,35,27,36);(5,8)(6,12)(7,16)(9,23)(10,21)(11,25)(13,28)(14,19)(15,29)(17,33)(18,32)(20,24)(22,27)(26,35)(30,36)(31,34)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261421);