Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #422711 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15309 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1067 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,8,4,7,11,6,19)(3,12,14,9,16,17,15,26)(5,20,24,31,23,10,22,18)(13,27,21,25,29,30,28,32);(1,3,8,14,7,16,6,15)(2,9,4,17,11,26,19,12)(5,21,24,29,23,28,22,13)(10,27,18,25,20,30,31,32);(1,4,6,11,7,19,8,2)(3,9,14,12,16,26,15,17)(5,18,22,20,23,31,24,10)(13,25,21,27,29,32,28,30);(1,5,6,22,7,23,8,24)(2,10,4,18,11,20,19,31)(3,13,14,21,16,29,15,28)(9,25,12,27,26,32,17,30)Polycyclic presentation:
2^9|a^2=ef,b^2=ef,c^2=e,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,f^c=fi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 329907);