Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25443 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#25445 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3060 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2656 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);(7,8,11,14,13,18,12,17)(9,19,20,15,23,30,22,31)(10,24,26,16,25,28,27,32)(21,29,34,37,36,38,35,33);(3,4)(5,6)(8,14,18,17)(9,20,23,22)(10,25)(16,24,32,28)(21,35,36,34)(26,27)(29,38)(33,37);(4,6)(7,9)(8,15)(10,21)(11,22)(12,20)(13,23)(14,19)(16,29)(17,30)(18,31)(24,37)(25,36)(26,35)(27,34)(28,33)(32,38);(7,10,12,27,13,25,11,26)(8,16,14,28,18,32,17,24)(9,21,20,34,23,36,22,35)(15,29,19,33,31,38,30,37)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351599);