Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#23744 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C24 | C22×D4 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2350 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);(3,4,7,9)(5,10,6,8)(11,12)(13,15)(14,16)(17,18);(3,5,7,6)(4,8,9,10)(11,13,14,17)(12,15,16,18)(19,20)(21,28)(22,26)(23,31)(24,27)(25,29)(30,33)(32,34);(3,6,7,5)(4,8,9,10)(12,16)(15,18)(19,21)(20,25)(22,29)(23,32)(24,30)(26,33)(27,34)(28,31);(19,22,24,23)(20,26,27,31)(21,29,30,32)(25,33,34,28)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339629);