Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×C4×#367 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | D8 |
| commutator | C8 | C22×C42 |
| Frattini, cosocle | C22×C8 | C24 |
| socle, prime radical | C24 | C2×D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
894 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(3,4,5,6);(7,8,9,10)(12,16)(13,20)(14,21)(17,22)(18,26)(19,25)(23,24);(11,12,17,26,14,18,13,16,15,19,20,24,21,23,22,25)176 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C4×#2177, C2×C8×#367, C2×#129146, … |
| maximal normal subgroup of | C2×C4×#2177, C2×C8×#367, C2×#129146, … |
| maximal quotient of | C2×C8×#367, C2×C4×#1426, C2×#151998 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400339);