Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10208 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#10209 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#3214 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1298 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)(5,12)(6,16)(7,10)(8,11)(9,13)(14,17)(15,18)(21,28)(22,23)(25,29)(26,32)(30,31)(33,34);(3,5,8,14)(4,9,11,18)(6,15,7,13)(10,17,16,12)(19,20)(21,29)(22,26)(23,32)(24,27)(25,31)(28,33)(30,34);(5,13)(6,7)(9,17)(10,16)(12,18)(14,15)(19,21)(20,25)(22,31)(23,28)(24,30)(26,34)(27,33)(29,32)50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 140011);