Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437793 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3006 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2740 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)(9,10,13,16,14,18,12,17)(11,15,21,19,22,23,20,24);(9,10)(11,19)(12,16)(13,17)(14,18)(15,21)(20,23)(22,24);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,20)(13,21)(14,22)(16,19)(17,24)(18,23);(1,4,5,3)(2,6,7,8)(9,11)(10,15)(12,20)(13,21)(14,22)(16,19)(17,24)(18,23);(9,12,14,13)(10,16,18,17)(11,20,22,21)(15,19,23,24)Polycyclic presentation:
2^9|a^2=f,c^2=g,d^2=g,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^b=hi,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, 6359225);