Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13398 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #481 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1874 |
| prime radical | C2×#469 | C4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
468 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 468 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) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,6,7,8)(9,10,12,20,15,21,23,18,16,19,24,17,14,11,13,22);(1,2,3,6,5,7,4,8)(9,11,15,19,16,21,14,10)(12,17,23,18,24,20,13,22);(1,3,5,4)(2,6,7,8)(9,12,14,13,16,24,15,23)(10,17,11,18,19,20,21,22)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,c^2=g,d^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,c^b=ch,d^b=dg,e^b=ei,f^b=fg,g^b=gi,h^b=hi,d^c=di,f^c=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43842);