Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23391 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1298 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 32 points:
(5,6,8,7)(9,10)(11,16)(12,14)(13,15)(17,18,21,23,22,25,20,26)(19,27,29,24,30,32,28,31);(1,2)(3,4)(9,11,13,12)(10,14,15,16)(18,23)(20,21)(24,27)(25,26)(28,29)(31,32);(2,4)(17,19,22,30)(18,24,25,31)(20,29,21,28)(23,27,26,32);(5,7,8,6)(9,12,13,11)(10,14,15,16)(18,25)(23,26)(24,31)(27,32)Polycyclic presentation:
2^9|a^2=h,b^2=efi,c^2=i,d^2=eh,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,e^b=ei,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266500);