Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9279 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#9288 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C24 | #121 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1092 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4,7,9,8,12,6,11)(5,13,15,10,16,17,14,18)(19,20,24,28,25,27,23,21)(22,29,30,26,31,34,32,33);(4,9)(6,7)(10,13)(11,12)(14,15)(17,18)(19,21,24,27,25,28,23,20)(22,29,30,33,31,34,32,26);(3,5,8,16)(4,10,12,18)(6,15,7,14)(9,13,11,17)(19,22,25,31)(20,26,27,33)(21,29,28,34)(23,32,24,30)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 106126);