Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #647656 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2918 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(6,8)(9,10)(11,15)(12,16)(13,14)(17,18,21,20)(19,23,24,22);(6,8)(10,14)(12,15)(17,19,21,24)(18,22,20,23);(5,6,7,8)(9,11,13,16)(10,15,14,12)(17,19)(18,23)(20,22)(21,24);(1,2,3,4);(9,12,13,15)(10,16,14,11)(17,20,21,18)(19,22,24,23)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=fh,d^2=i,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 582202);