Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12028 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C2×#13185 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#106 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C24 | C2×SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
736 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2);(4,6)(7,8,10,15,12,16,11,21)(9,17,19,22,20,13,18,14);(8,13,16,17)(9,18)(10,11)(14,15,22,21)(19,20);(3,4,5,6)92 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#495476, C2×#495843, C2×#496007, … |
| maximal normal subgroup of | C2×#495476, C2×#495843, C2×#496007, … |
| maximal quotient of | C2×#140388, C2×#147283, #72309900, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402267);