Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437999 |
| 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 | C24 | C25 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
3750 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)(11,14)(12,20)(13,16)(15,22)(17,19)(18,24)(21,23);(9,10)(11,17,19,14)(12,15)(13,16)(18,24,21,23)(20,22);(1,3,5,4)(2,6,7,8)(9,11)(10,14)(12,21)(13,19)(15,24)(16,17)(18,22)(20,23);(1,4,5,3)(2,6,7,8)(9,11)(10,14)(12,18)(13,19)(15,23)(16,17)(20,24)(21,22);(9,12,13,22)(10,15,16,20)(11,18,19,21)(14,23,17,24)Polycyclic presentation:
2^9|b^2=f,c^2=g,d^2=g,e^2=i,b^a=bf,c^a=cg,e^a=ei,h^a=hi,c^b=ch,d^b=dh,h^b=hi,e^c=ei,f^c=fi,f^d=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6359471);