Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C8 · Schur multiplier (H2) = C29 |
| direct product | C22×C8×Q16 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C8 | D4 |
| commutator | C4 | C24×C8 |
| Frattini, cosocle | C42 | C25 |
| socle, prime radical | C24 | C4×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2138 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2);(3,4);(5,6,7,8,9,10,11,12);(13,14,15,16,17,18,19,20)(21,28,27,26,25,24,23,22);(13,21,17,25)(14,22,18,26)(15,23,19,27)(16,24,20,28)224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6251703);