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 | #34937 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2708 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(5,6,8,7)(9,10,15,17,14,18,13,20)(11,19,16,23,21,12,22,24);(6,7)(9,11,13,22,14,21,15,16)(10,12,20,23,18,19,17,24);(2,4)(9,12)(10,16)(11,20)(13,23)(14,19)(15,24)(17,21)(18,22);(5,7,8,6)(9,13,14,15)(10,17,18,20)(11,16,21,22)(12,23,19,24);(11,21)(12,19)(16,22)(23,24)Polycyclic presentation:
2^9|a^2=h,b^2=fh,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=di,d^b=df,e^b=ei,e^c=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6289015);