Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #616960 |
| 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 | |
3108 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)(11,13)(12,15)(14,16)(17,18,20,22)(19,21,23,24);(10,11)(14,15)(17,19,20,23)(18,21,22,24);(5,6,7,8)(17,19)(18,21)(20,23)(22,24);(1,2)(3,4)(9,11,13,10)(12,15,16,14)(17,18)(19,21)(20,22)(23,24);(9,12)(10,14)(11,15)(13,16)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=g,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=df92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 524738);