Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C8 · Schur multiplier (H2) = C211 |
| direct product | C22×#1797 |
| subgroup | G/H | |
|---|---|---|
| center | C24×C4 | C23 |
| commutator | C22 | C24×C8 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C25 | C22×C4 |
| prime radical | C26 | C2×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
8242 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2);(3,4);(6,8)(9,10)(11,12)(13,14,15,17,16,18,19,20);(5,6,7,8);(10,12)224 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#392160, C22×#32956, C2×#392954, … |
| maximal normal subgroup of | C2×#392160, C22×#32956, C2×#392954, … |
| maximal quotient of | C22×#8595 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6277955);