Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#25191 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3012 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3182 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 38 points:
(1,2);(7,8,12,19,14,20,13,18)(9,21,25,35,26,34,24,15)(10,27,30,17,11,31,29,16)(22,32,23,33,38,37,28,36);(3,4)(5,6)(7,9,12,25,14,26,13,24)(8,15,19,21,20,35,18,34)(10,28,30,22,11,23,29,38)(16,36,27,32,17,33,31,37);(4,6)(7,10,14,11)(8,16,20,17)(9,22,26,38)(12,29,13,30)(15,32,35,37)(18,27,19,31)(21,36,34,33)(23,25,28,24);(7,11,14,10)(8,17,20,16)(9,23,26,28)(12,29,13,30)(15,33,35,36)(18,27,19,31)(21,37,34,32)(22,25,38,24)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346211);