Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6342866 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×C22⋊C4 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3549 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3542 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10,16,19,15,20,14,12)(11,21,13,18,22,17,23,24);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,18)(13,16)(14,23)(15,22)(19,24)(20,21);(9,12,16,20,15,19,14,10)(11,18,13,21,22,24,23,17);(3,8)(5,7)(9,13)(10,18)(11,14)(12,21)(15,23)(16,22)(17,19)(20,24);(2,6)(5,7)(9,14,15,16)(10,19,20,12)(11,13,22,23)(17,18,21,24)Polycyclic presentation:
2^9|a^2=h,c^2=gh,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,e^b=ei,g^c=gi,h^c=hi,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6300746);