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 | C2×#135 | 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 | |
319 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 319 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)(3,6)(7,8)(10,16)(11,17,21,19)(12,15)(13,18)(20,22,23,24);(1,2,6,8,4,7,3,5)(9,10,12,16,14,18,15,13)(11,19,20,22,21,17,23,24);(2,7)(5,8)(9,11)(10,17)(12,20)(13,22)(14,21)(15,23)(16,24)(18,19)Polycyclic presentation:
2^9|a^2=f,b^2=dgi,d^2=g,e^2=h,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 | #72309931, #72309938, #72309939, … |
| maximal normal subgroup of | #72309931, #72309938, #72309939, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42230);