Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9331 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#9332 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3194 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1268 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);(3,4,9,12)(5,8,11,7)(6,13)(10,14)(15,17)(16,18)(19,20)(21,24,28,32)(22,29,30,25)(23,26)(27,31)(33,34);(3,5,7,4,9,11,8,12)(6,13,15,10,16,18,14,17)(19,21,23,28)(20,24)(22,27,30,33)(25,31)(26,32)(29,34);(3,6)(4,10)(5,13)(7,15)(8,14)(9,16)(11,18)(12,17)(19,22)(20,25)(21,27)(23,30)(24,31)(26,29)(28,33)(32,34)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 106713);