Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30428 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#30870 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2557 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);(7,8,13,18)(9,19,22,33)(10,23,26,15)(11,17,12,16)(14,27,30,21)(20,31,36,37)(24,29,35,38)(25,32,28,34);(3,4)(5,6)(7,9,13,22)(8,14,18,30)(10,24,26,35)(11,27,12,21)(15,31,23,37)(16,33,17,19)(20,28,36,25)(29,34,38,32);(4,6)(7,9)(8,14)(10,20)(11,21)(12,27)(13,22)(15,29)(16,19)(17,33)(18,30)(23,38)(24,25)(26,36)(28,35)(31,32)(34,37);(7,10,12,28,13,26,11,25)(8,15,17,34,18,23,16,32)(9,20,27,35,22,36,21,24)(14,29,33,37,30,38,19,31)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7409343);