Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24292 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C2×#24293 |
| 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 | |
4448 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);(3,4)(5,6)(7,8)(9,15)(10,23)(11,26)(12,19)(13,18)(14,21)(16,27)(17,24)(20,35)(22,30)(25,29)(28,34)(31,36)(32,37)(33,38);(7,9,12,21,13,10,24,22)(8,14,17,15,18,30,19,23)(11,20,27,36,28,25,38,37)(16,29,26,31,33,35,34,32);(4,6)(7,10)(8,15)(9,13)(11,20)(12,22)(14,19)(16,32)(17,30)(18,23)(21,24)(25,28)(26,29)(27,36)(31,33)(34,35)(37,38);(7,11,24,38,13,28,12,27)(8,16,19,34,18,33,17,26)(9,20,22,37,10,25,21,36)(14,29,23,32,30,35,15,31)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6349904);