Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C25 | C2×D4 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
2430 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(6,8)(13,14,15,16)(18,22)(19,20)(23,24);(2,4)(9,10)(11,12)(17,18,20,23,21,24,19,22);(5,6,7,8)(10,12)(18,23)(19,20)(22,24)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,d^2=i,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,h^a=hi,c^b=cf,d^c=di,f^c=fi,h^c=hi128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31156);