Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C43 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#7845 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C42 | C23 |
| commutator | C22 | C2×C43 |
| Frattini, cosocle | C2×C42 | C24 |
| socle, prime radical | C24 | C2×C42 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1202 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 38 points:
(1,2);(7,8,10,14,11,13,17,20)(9,12,16,18,15,19,21,22)(23,24)(25,28)(26,30)(27,29)(31,35)(32,34)(33,36)(37,38);(7,9)(8,12)(10,16)(11,15)(13,19)(14,18)(17,21)(20,22)(23,25,26,31,27,32,33,37)(24,28,30,35,29,34,36,38);(3,4,5,6)(8,13)(9,15)(14,20)(16,21)(24,29)(28,34)(30,36)(35,38)224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 87995);