Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23391 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1382 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 24 points:
(5,6,8,7)(9,10)(11,15)(12,14)(13,16)(17,18,21,24)(19,22,20,23);(1,2)(3,4)(9,11,13,12)(10,14,16,15)(17,19,21,20)(18,22,24,23);(2,4)(9,11,13,12)(10,15,16,14)(18,23)(19,20)(22,24);(5,7,8,6)(9,12,13,11)(10,14,16,15)(17,19,21,20)(18,22,24,23)Polycyclic presentation:
2^9|a^2=h,b^2=ei,c^2=e,d^2=eh,f^2=i,b^a=be,c^a=cf,f^a=fi,c^b=cg,d^c=di,f^c=fi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266414);