Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #7855 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C42 |
| commutator | C2×C4 | C43 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C23 | C2×#97 |
| prime radical | C24 | C2×M16 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
632 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)(3,4)(9,10,12,11,14,16,22,18,15,17,13,19,21,20,23,24);(2,4)(5,6,7,8);(9,11,13,16,14,18,23,17,15,19,12,20,21,24,22,10)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=g,e^2=h,g^2=h,h^2=i,b^a=bd,c^a=ch,g^a=gi,e^c=ei128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53517);