Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438880 |
| 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 |
| prime radical | C2×#3038 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2118 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,7)(3,6,4,8)(9,10,16,19,15,11,14,20)(12,21,24,22,23,17,13,18);(1,3,5,4)(2,6,7,8)(9,11,15,10)(12,21,23,17)(13,22,24,18)(14,20,16,19);(1,4,5,3)(2,6,7,8)(9,12,16,24,15,23,14,13)(10,17,19,18,11,21,20,22);(9,13,14,23,15,24,16,12)(10,18,20,17,11,22,19,21);(9,14,15,16)(10,19,11,20)(12,13,23,24)(17,18,21,22)Polycyclic presentation:
2^9|a^2=fg,b^2=g,c^2=fg,d^2=f,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,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, 6355272);