Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10496 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#10499 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #394 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C24 | #123 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
828 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);(3,4,8,12)(5,6,10,16)(7,13,18,11)(9,17,15,14)(19,20)(21,24)(22,25)(23,27)(26,31)(28,32)(29,34)(30,33);(3,5)(4,9)(6,13)(7,17)(8,10)(11,16)(12,15)(14,18)(19,21,23,29)(20,24,27,34)(22,30,31,28)(25,33,26,32);(3,6,7,15,8,16,18,9)(4,10,11,17,12,5,13,14)(20,25)(22,31)(24,32)(26,27)(28,30)(33,34)62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 147867);