Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#21670 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C24 | C23×C4 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1772 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4)(5,10)(6,8)(7,9)(11,12)(13,15)(14,16)(17,18)(19,20,23,22,24,26,31,30)(21,25,29,28,27,32,34,33);(3,5,7,6)(4,8,9,10)(11,13,14,17)(12,15,16,18)(19,21)(20,25)(22,28)(23,29)(24,27)(26,32)(30,33)(31,34);(3,6,7,5)(4,8,9,10)(12,16)(15,18)(21,27)(25,32)(28,33)(29,34);(19,22,23,26,24,30,31,20)(21,28,29,32,27,33,34,25)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6280750);