Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C214×C4 |
| direct product | C22×C4×#127 |
| subgroup | G/H | |
|---|---|---|
| center | C24×C4 | C23 |
| commutator | C22 | C25×C4 |
| Frattini, cosocle | C23 | C26 |
| socle | C25 | C24 |
| prime radical | C23×#127 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
23178 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 16 points:
(1,2);(3,4);(5,6,7,8);(10,12)(14,15);(13,14,16,15);(9,10,11,12)(13,15,16,14)224 conjugacy classes — view the full character table.
| maximal subgroup of | C22×#9644, C22×#23798, C2×C42×#127, … |
| maximal normal subgroup of | C22×#9644, C22×#23798, C2×C42×#127, … |
| maximal quotient of | C2×C42×#127, C2×C4×#2397, C22×#21690, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532452);