Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1828 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C29 |
| direct product | C2×#10770 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #381 |
| commutator | C23 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4010 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 38 points:
(1,2);(3,4)(5,6)(7,8,11,16,12,15,25,30)(9,13,19,28,18,27,23,31)(10,20,21,32,22,14,24,29)(17,33,35,38,34,26,36,37);(7,9)(8,13,15,27)(10,17)(11,23)(12,18)(14,26,20,33)(16,31,30,28)(19,25)(21,36)(22,34)(24,35)(29,38,32,37);(4,6)(7,10,12,22)(8,14,15,20)(9,17,18,34)(11,24,25,21)(13,26,27,33)(16,32,30,29)(19,36,23,35)(28,38,31,37)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260544);