Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24802 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×#28498 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3062 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 38 points:
(1,2);(7,8,14,18,13,20,12,19)(9,17,11,30,23,32,22,15)(10,24,27,16,29,36,28,35)(21,33,37,34,25,31,26,38);(3,4,5,6)(7,9,12,22,13,23,14,11)(8,15,18,17,20,30,19,32)(10,25,27,26,29,21,28,37)(16,33,24,38,35,31,36,34);(4,6)(7,10)(8,16)(9,21)(11,26)(12,27)(13,29)(14,28)(15,31)(17,34)(18,24)(19,36)(20,35)(22,37)(23,25)(30,33)(32,38);(7,11,14,23,13,22,12,9)(8,17,19,15,20,32,18,30)(10,26,28,25,29,37,27,21)(16,34,36,31,35,38,24,33)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6372782);