Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C2×#24290 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4458 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)(5,6)(7,8)(9,15)(10,23)(11,26)(12,19)(13,18)(14,21)(16,27)(17,24)(20,34)(22,30)(25,29)(28,33)(31,35)(32,38)(36,37);(7,9,12,21,13,10,24,22)(8,14,17,15,18,30,19,23)(11,20,27,35,28,25,38,36)(16,29,26,31,32,34,33,37);(4,6)(7,10)(8,15)(9,13)(11,25)(12,22)(14,19)(16,31)(17,30)(18,23)(20,28)(21,24)(26,34)(27,36)(29,33)(32,37)(35,38);(7,11,24,38,13,28,12,27)(8,16,19,33,18,32,17,26)(9,20,22,36,10,25,21,35)(14,29,23,37,30,34,15,31)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6349900);