Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29592 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#29645 |
| 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 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3242 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);(7,8,11,14)(9,15,17,12)(10,13,18,20)(16,21,22,19)(23,24)(25,32)(26,30)(27,35)(28,31)(29,33)(34,37)(36,38);(3,4)(5,6)(7,9,11,17)(8,12,14,15)(10,16,18,22)(13,19,20,21);(4,6)(7,10)(8,13)(9,16)(11,18)(12,19)(14,20)(15,21)(17,22)(23,25,28,34)(24,29,31,38)(26,33,27,36)(30,37,35,32);(10,18)(13,20)(16,22)(19,21)(23,26,28,27)(24,30,31,35)(25,33,34,36)(29,37,38,32)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6341863);