Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C16 · Schur multiplier (H2) = C28 |
| direct product | C22×#1374 |
| subgroup | G/H | |
|---|---|---|
| center | C24×C8 | C22 |
| commutator | C2 | C22×C4×C16 |
| Frattini, cosocle | C22×C8 | C24 |
| socle, prime radical | C25 | C2×C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
2146 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2);(3,4);(5,6)(7,8)(13,14,15,18,16,19,21,24,17,20,22,25,23,26,27,28);(6,8)(9,10,11,12)320 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C22×#12599, C22×#12937, C22×#13012 |
| maximal normal subgroup of | C22×#12599, C22×#12937, C22×#13012 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400031);