Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7396 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #353 |
| commutator | C2×M16 | C2×C8 |
| Frattini, cosocle | #1728 | C22 |
| socle | C22 | #1352 |
| prime radical | C24 | #103 |
| LCS | G ▷ C2×M16 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C23×C4 ◁ #1728 ◁ G | |
| LECS | G ▷ #1728 ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1 | |
256 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 256 conjugacy classes of subgroups — too many to draw. Only its 48 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 — 2 generators on 24 points:
(1,2,3,5,4,6,7,8)(10,14)(11,18,21,22)(15,24)(16,19)(17,23);(9,10,12,16,13,17,20,15)(11,19,18,14,21,24,22,23)Polycyclic presentation:
2^9|a^2=d,b^2=eh,c^2=fhi,d^2=g,e^2=i,f^2=i,b^a=bc,c^a=ce,e^a=eh,c^b=cf,d^b=defhi,e^b=ei,h^b=hi,d^c=dhi,e^c=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1149);