Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29736 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#29881 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1952 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)(9,13)(10,14)(11,12)(15,21)(16,19)(17,22)(18,20)(23,24,28,30,27,32,26,31)(25,29,35,36,33,37,34,38);(3,4)(5,6)(8,12)(13,19)(14,20)(21,22)(23,25)(24,29)(26,34)(27,33)(28,35)(30,36)(31,38)(32,37);(4,6)(7,9,11,16)(8,13,12,19)(10,17,18,15)(14,22,20,21)(24,30)(26,28)(29,36)(31,32)(34,35)(37,38);(7,10,11,18)(8,14,12,20)(9,15,16,17)(13,21,19,22)(25,33)(29,37)(34,35)(36,38)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6343378);