Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9263 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#9263 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(4,8)(5,11)(6,14)(10,16)(12,13)(15,17)(19,20)(21,24,28,32)(22,29,30,25)(23,26)(27,31)(33,34);(3,4,6,8,7,10,14,16)(5,9,12,15,13,18,11,17)(19,21)(20,24,26,32)(22,27)(23,28)(25,31,29,34)(30,33);(3,5)(4,9)(6,12)(7,13)(8,15)(10,18)(11,14)(16,17)(19,22)(20,25)(21,27)(23,30)(24,31)(26,29)(28,33)(32,34)86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 105688);