Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10488 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#10495 |
| 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 | |
886 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)(5,8)(6,9)(7,11)(10,15)(12,16)(13,18)(14,17)(19,20,24,28)(21,29,30,33)(22,25,23,27)(26,31,34,32);(3,5,7,13)(4,8,11,18)(6,14,15,12)(9,17,10,16)(20,25)(21,30)(22,23)(26,29)(27,28)(33,34);(4,9)(6,15)(8,16)(10,11)(12,14)(17,18)(19,21,22,31,24,30,23,32)(20,26,27,33,28,34,25,29)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 147751);