Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29742 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×#30326 |
| 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 | |
2174 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)(5,8)(6,7)(9,10)(11,12)(13,16)(14,15)(17,18)(21,27)(24,32)(28,34)(29,31);(4,7)(8,10)(19,20,23,26,22,25,30,33)(21,28,29,32,27,34,31,24);(3,5)(4,8)(6,9)(7,10)(12,15)(16,18)(19,21,22,27)(20,24,25,32)(23,31,30,29)(26,34,33,28);(5,9)(8,10)(11,13,14,17)(12,16,15,18)(20,25)(24,32)(26,33)(28,34)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6341057);