Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
| direct product | C4×#2634 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator | C23 | C24×C4 |
| Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2590 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,3,4);(9,10)(11,14)(12,13)(15,16)(17,18,21,24)(19,25,27,22)(20,23,28,30)(26,31,32,29);(5,6)(7,8)(17,19,21,27)(18,22,24,25)(20,26,28,32)(23,29,30,31);(6,8)(10,13)(14,16)(17,20,21,28)(18,23,24,30)(19,26,27,32)(22,29,25,31);(9,11,12,15)(10,14,13,16)(20,28)(23,30)(26,32)(29,31)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 422396);