Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511751 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
3261 conjugacy classes of subgroups; only the 224 normal subgroups are shown.
This group has 3261 conjugacy classes of subgroups — too many to draw. Only its 224 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(6,10)(7,13)(8,16)(11,18)(12,19)(14,17)(15,20);(5,6)(7,14)(8,17)(9,11)(10,13)(12,16)(15,19)(18,20);(7,15)(10,18)(13,20)(14,19);(2,4)Polycyclic presentation:
2^9|e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,h^a=hi,e^b=eh,f^b=fi,h^b=hi,e^c=ei62 conjugacy classes — view the full character table.
| maximal subgroup of | #72224588, #72249968, #72250058, … |
| maximal normal subgroup of | #72224588, #72249968, #72250058, … |
| maximal quotient of | #72224569 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 418947);