Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29592 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×#29915 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1994 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);(3,4,6,7)(5,8,9,10)(11,12)(13,15)(14,16)(17,18)(20,25)(26,33)(27,32)(28,34);(4,7)(8,10)(11,13,14,17)(12,15,16,18)(19,20)(21,28)(22,27)(23,25)(24,32)(26,31)(29,33)(30,34);(3,5)(4,8)(6,9)(7,10)(12,16)(15,18)(19,21,23,30)(20,26,25,33)(22,31,24,29)(27,34,32,28);(5,9)(8,10)(19,22,23,24)(20,27,25,32)(21,29,30,31)(26,28,33,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6341283);