Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #52096 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2719 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(5,6)(7,8)(9,10,14,19,15,18,20,24)(11,16,12,17,21,13,22,23);(1,2)(3,4)(9,11)(10,16)(12,14)(13,18)(15,21)(17,19)(20,22)(23,24);(6,8)(9,12)(10,17)(11,20)(13,19)(14,21)(15,22)(16,24)(18,23);(2,4)(10,18)(11,21)(12,22)(19,24);(9,13,14,23,15,16,20,17)(10,12,19,21,18,22,24,11)Polycyclic presentation:
2^9|a^2=h,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ci,d^b=dfi,e^b=ei,d^c=di116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6290980);