Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9369 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#9371 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#3214 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1194 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4,7,11,8,10,6,16)(5,12,15,17,14,18,13,9)(19,20,24,26,23,22,28,31)(21,27,29,34,30,25,32,33);(3,5)(4,9,10,17)(6,13)(7,15)(8,14)(11,12,16,18)(19,21,28,29,23,30,24,32)(20,25,26,34,22,27,31,33);(4,10)(6,7)(9,17)(13,15)(19,22)(20,23)(21,27)(24,26)(25,30)(28,31)(29,33)(32,34)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 107360);