Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #686200 |
| 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 | #14811 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3008 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)(9,10,13,16)(11,12,15,14)(17,18)(19,22)(20,21)(23,24);(6,8)(9,11,13,15)(10,14,16,12)(17,19,23,21)(18,20,24,22);(5,6)(7,8)(9,12,13,14)(10,15,16,11)(17,20,23,22)(18,19,24,21);(1,2,3,4)(9,12)(10,11)(13,14)(15,16)(17,21,23,19)(18,20,24,22);(9,11)(10,12)(13,15)(14,16)(17,22,23,20)(18,19,24,21)Polycyclic presentation:
2^9|a^2=fgh,b^2=f,c^2=f,d^2=fi,e^2=gh,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 628827);