Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C216 |
| direct product | C22×D4×C22⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C24 |
| commutator | C22 | C25×C4 |
| Frattini, cosocle | C23 | C26 |
| prime radical | C25×D4 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
41962 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 16 points:
(1,2);(3,4);(5,6,7,8);(6,8);(9,10,12,14)(11,13,15,16);(10,13)(14,16)200 conjugacy classes — view the full character table.
| maximal subgroup of | C22×#35680, #72231722, C22×D4×#120, … |
| maximal normal subgroup of | C22×#35680, #72231722, C22×D4×#120, … |
| maximal quotient of | C2×D4×#407, C2×D4×#418, #72233563, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532523);