Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9239 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#9242 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1304 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)(5,8)(6,14,17,12)(7,13)(9,11)(10,16,18,15)(19,20)(21,24,28,31)(22,25)(23,26)(27,30,33,34)(29,32);(3,5,8,4,9,13,7,11)(6,12,15,10,17,14,16,18)(19,21,23,28)(20,24)(22,27,29,33)(25,30)(26,31)(32,34);(3,6,9,17)(4,10,11,18)(5,12,13,14)(7,16,8,15)(19,22,23,29)(20,25,26,32)(21,27,28,33)(24,30,31,34)86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 105307);