Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3044 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(5,6,8,7)(9,10,14,12)(11,16,13,15)(18,22)(20,23);(1,2,3,4)(5,7,8,6)(9,11,14,13)(10,15,12,16)(17,18,21,22)(19,23,24,20);(1,2)(3,4)(10,12)(15,16)(18,22)(19,24);(9,12,14,10)(11,16,13,15)(17,19)(18,23)(20,22)(21,24);(9,13,14,11)(10,16,12,15)(17,20,21,23)(18,19,22,24)Polycyclic presentation:
2^9|a^2=i,b^2=hi,d^2=g,e^2=f,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^c=df128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 423820);