Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #135024 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1802 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C22×D8 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
329 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 329 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,5)(3,7)(6,8)(9,10,12,16,13,18,23,17)(11,19,21,15,22,14,20,24);(1,2,3,5,4,6,7,8);(2,6)(5,8)(10,14)(11,20)(12,23)(15,16)(17,24)(18,19)(21,22)Polycyclic presentation:
2^9|a^2=f,b^2=d,d^2=g,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=cg,e^c=ehi,f^c=fi,h^c=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72342433, #72342438, #72342439, … |
| maximal normal subgroup of | #72342433, #72342438, #72342439, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42222);