Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470811 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2150 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)(3,5)(4,6)(7,8)(10,15)(11,18)(12,13)(16,24)(17,22)(19,23);(1,3)(2,5)(4,7)(6,8)(9,10,12,22,14,17,13,15)(11,19,20,23,18,16,21,24);(3,7)(5,8)(9,11,13,21,14,18,12,20)(10,16,15,23,17,19,22,24);(3,7)(5,8)(10,17)(15,22)(16,19)(23,24);(2,6)(5,8)(9,12,14,13)(10,15,17,22)(11,20,18,21)(16,23,19,24)Polycyclic presentation:
2^9|b^2=fi,c^2=f,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,e^b=ei,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6378010);