Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C8 · Schur multiplier (H2) = C210 |
| direct product | C22×C8×D8 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C8 | D4 |
| commutator | C4 | C24×C8 |
| Frattini, cosocle | C42 | C25 |
| socle | C24 | C4×D4 |
| prime radical | C23×D8 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2746 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);(5,6,7,8,9,10,11,12);(13,14,15,16,17,18,19,20);(14,20)(15,19)(16,18)224 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C4×C8×D8, C22×C8×D16, C2×C8×#357, … |
| maximal normal subgroup of | C2×C4×C8×D8, C22×C8×D16, C2×C8×#357, … |
| maximal quotient of | C2×C4×C8×D8, C22×C8×D16, C22×#12045, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6251701);