Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #142404 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1616 |
| commutator, Frattini, cosocle | C22×M16 | C23 |
| prime radical | C2×#469 | C22 |
| LCS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
417 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 417 conjugacy classes of subgroups — too many to draw. Only its 70 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 — 3 generators on 20 points:
(1,2)(3,4)(6,10)(7,16)(8,11)(12,13)(14,17)(15,19)(18,20);(5,6,11,14,9,15,8,13)(7,17,20,10,18,12,16,19);(2,4)(5,7,8,16,9,18,11,20)(6,12,14,19,15,17,13,10)Polycyclic presentation:
2^9|b^2=gh,c^2=g,d^2=gi,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fhi,g^a=gi,c^b=cf,d^b=dgi,g^b=gi,h^b=hi,d^c=dh,f^c=fi,f^d=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72250333, #72250343, #72250346, … |
| maximal normal subgroup of | #72250333, #72250343, #72250346, … |
| maximal quotient of | #72265323, #72265332 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 49600);