Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437935 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3390 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)(3,8)(4,6)(5,7)(9,10)(11,16)(12,17)(13,19)(14,18)(15,20)(21,23)(22,24);(9,11,12,20,13,21,14,22)(10,15,18,16,19,24,17,23);(1,3,5,4)(2,6,7,8)(9,10,13,19)(11,16,21,23)(12,18,14,17)(15,20,24,22);(1,4,5,3)(2,6,7,8)(9,11,13,21)(10,16,19,23)(12,20,14,22)(15,18,24,17);(9,12,13,14)(10,17,19,18)(11,20,21,22)(15,23,24,16)Polycyclic presentation:
2^9|b^2=f,c^2=g,d^2=g,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,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, 6359428);