Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2756 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)(6,8)(9,10,15,11,14,17,12,19)(13,16,22,18,21,23,20,24);(2,4)(9,11,15,17,14,19,12,10)(13,18,22,23,21,24,20,16);(5,6,7,8)(9,12,14,15)(10,11,17,19)(13,20,21,22)(16,18,23,24);(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24);(13,21)(16,23)(18,24)(20,22)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=g,h^2=i,b^a=bf,c^a=cg,c^b=ci,e^d=ei128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6279113);