Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C211 |
| direct product | C2×#23891 |
| 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 | |
5897 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)(7,8,12,17)(9,18,22,19)(10,24,26,32)(11,27,13,16)(14,23,30,21)(15,28,31,25)(20,29,36,38)(33,37,34,35);(7,9,12,22)(8,14,17,30)(10,20,26,36)(11,23,13,21)(15,29,31,38)(16,19,27,18)(24,33,32,34)(25,37,28,35);(4,6)(7,8)(9,19)(10,15)(11,16)(12,17)(13,27)(14,21)(18,22)(20,34)(23,30)(24,25)(26,31)(28,32)(29,37)(33,36)(35,38);(7,10,12,26)(8,15,17,31)(9,20,22,36)(11,28,13,25)(14,29,30,38)(16,32,27,24)(18,33,19,34)(21,37,23,35)86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344796);