Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×C22⋊C4 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C2×#336 |
| prime radical | C24 | #97 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
471 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,3,6,4,5,7,8)(9,10,13,19,16,18,23,24,15,17,22,21,12,11,14,20);(2,5)(3,7)(9,11,14,21,12,17,22,24,15,18,23,19,16,10,13,20);(10,17)(11,18)(19,21)(20,24)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=i,e^2=h,f^2=gh,h^2=i,b^a=bd,c^a=ci,d^a=dg,c^b=ci,d^b=di,e^b=egi128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34362);