Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6439029 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #60369 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2663 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)(3,6)(4,5)(7,8)(9,10)(11,21)(12,14)(13,18)(15,19)(16,20)(17,22)(23,24);(2,5)(6,8)(9,11,15,23,16,13,14,22)(10,17,12,18,20,24,19,21);(1,3,4,7)(2,6,5,8)(9,12,14,10,16,19,15,20)(11,17,22,21,13,24,23,18);(9,13)(10,18)(11,16)(12,24)(14,23)(15,22)(17,19)(20,21);(9,14,16,15)(10,12,20,19)(11,22,13,23)(17,18,24,21)Polycyclic presentation:
2^9|b^2=g,c^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,h^b=hi,e^c=ei,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6348042);