Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #97058 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #379 |
| commutator, prime radical | C25 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1024 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 1024 conjugacy classes of subgroups — too many to draw. Only its 162 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:
(9,10,11,14,12,13,15,16)(21,24);(1,2,4,6)(3,7,8,5)(10,13)(11,15)(17,18)(19,21)(20,22)(23,24);(1,3,4,8)(2,5,6,7)(17,19)(18,21)(20,23)(22,24)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=f,g^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=cf,e^b=ei,g^b=gh,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 29242);