Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24278 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C2×#24280 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4588 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)(9,17)(10,24)(11,28)(12,19)(13,18)(14,20)(15,22)(16,26)(21,35)(23,31)(25,38)(27,34)(29,33)(30,37)(32,36);(3,4)(5,6)(7,9,12,22,14,23,13,11)(8,15,18,17,20,28,19,31)(10,21,26,36,27,37,29,25)(16,30,24,32,33,35,34,38);(4,6)(7,10)(8,16)(9,21)(11,25)(12,26)(13,29)(14,27)(15,30)(17,32)(18,24)(19,34)(20,33)(22,36)(23,37)(28,35)(31,38);(7,11,13,23,14,22,12,9)(8,17,19,15,20,31,18,28)(10,25,29,37,27,36,26,21)(16,32,34,30,33,38,24,35)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6349774);