Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32944 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C2×#32946 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4482 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)(7,8,13,18)(9,19,21,30)(10,24,26,32)(11,17,12,16)(14,23,29,22)(15,27,31,25)(20,34,35,28)(33,37,38,36);(7,9,12,23,13,21,11,22)(8,14,17,30,18,29,16,19)(10,20,27,37,26,35,25,36)(15,28,32,38,31,34,24,33);(7,10)(8,15)(9,20)(11,25)(12,27)(13,26)(14,28)(16,24)(17,32)(18,31)(19,33)(21,35)(22,36)(23,37)(29,34)(30,38);(4,6)(9,21)(14,29)(19,30)(20,35)(22,23)(28,34)(33,38)(36,37)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536226);