Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10645 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
| direct product | C2×#10645 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #397 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
628 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);(3,4,6,11,8,12,7,10)(5,9,14,17,15,16,13,18)(19,20,22,27,24,28,23,25)(21,29,30,34,32,33,31,26);(3,5)(4,9)(6,14)(7,13)(8,15)(10,18)(11,17)(12,16)(20,25)(21,30)(22,23)(27,28)(29,33)(31,32);(4,10)(5,13)(6,7)(9,16)(11,12)(14,15)(19,21,22,31,24,32,23,30)(20,26,27,33,28,34,25,29)62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 150284);