Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30490 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×#31075 |
| 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 | |
3448 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)(9,18,22,29)(10,15)(11,16)(12,17)(13,20,14,21)(19,31,38,36)(23,30,37,28)(24,32)(25,33)(26,34)(27,35);(3,4)(5,6)(7,9)(8,13)(10,23)(11,21)(12,22)(14,17)(15,31)(16,29)(18,35)(19,24)(20,27)(25,37)(26,38)(28,32)(30,34)(33,36);(4,6)(7,9)(8,14)(10,19)(11,20)(12,22)(13,17)(15,30)(16,29)(18,35)(21,27)(23,24)(25,38)(26,37)(28,33)(31,34)(32,36);(7,10,12,25)(8,15,17,33)(9,19,22,38)(11,26,27,24)(13,28,14,30)(16,34,35,32)(18,36,29,31)(20,37,21,23)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410561);