Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| 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 | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2624 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)(9,10,13,16)(11,15,12,14)(17,18)(19,23)(20,22)(21,24);(6,8)(11,12)(14,15)(17,18,20,22)(19,23,24,21);(5,6)(7,8)(9,11,13,12)(10,14,16,15)(17,18,20,22)(19,21,24,23);(1,2,3,4)(9,12,13,11)(10,15,16,14)(17,18)(19,21)(20,22)(23,24);(9,10)(11,14)(12,15)(13,16)(17,19,20,24)(18,21,22,23)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=g,d^2=i,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1520866);