Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1366 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C22×#1366 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #333 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| prime radical | C24 | #100 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
758 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);(5,6,8,13,21,14,32,30,10,15,23,31,9,12,22,28)(7,16,18,29,24,27,20,35,17,33,34,36,19,11,25,26);(5,7,10,17)(6,11)(8,20,23,25)(9,24,21,19)(12,16)(13,29)(14,33)(15,27)(18,22,34,32)(26,30)(28,35)(31,36)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390295);