Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12173 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1598 |
| commutator | C2×M16 | C22×C4 |
| Frattini, cosocle | C22×M16 | C23 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
382 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 382 conjugacy classes of subgroups — too many to draw. Only its 86 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:
(1,2,4,3)(5,6)(7,11)(8,12)(9,14)(10,15)(13,16)(17,20)(18,19);(1,3,4,2)(6,11)(7,16)(8,9)(12,13)(14,19)(15,20)(17,18);(6,12,15,14)(11,13,20,19)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=h,d^2=gi,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,h^a=hi,d^b=dg,e^b=ehi,g^b=gi,d^c=dh,e^d=ei50 conjugacy classes — view the full character table.
| maximal subgroup of | #72209821 |
| maximal normal subgroup of | #72209821 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43245);