Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3288 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
| direct product | C2×#23203 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2678 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);(3,4,6,5)(7,8,13,18,14,19,12,17)(9,20,24,15,25,32,23,33)(10,26,27,35,28,34,11,16)(21,37,29,36,22,30,38,31);(3,5,6,4)(7,9,14,25)(8,15,19,33)(10,21,28,22)(11,29,27,38)(12,24,13,23)(16,30,35,37)(17,32,18,20)(26,36,34,31);(7,10,12,11,14,28,13,27)(8,16,18,26,19,35,17,34)(9,21,24,29,25,22,23,38)(15,30,20,36,33,37,32,31);(7,11,12,28,14,27,13,10)(8,16,17,34,19,35,18,26)(9,22,23,29,25,21,24,38)(15,31,20,30,33,36,32,37)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6290006);