Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393909 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2678 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,3,4)(5,6,7,8)(17,18,20,25,23,28,22,27)(19,29,31,24,21,32,30,26);(2,4)(9,10)(11,14)(12,13)(15,16)(17,19,23,21)(18,24,28,26)(20,30,22,31)(25,29,27,32);(6,8)(10,13)(14,16)(17,20,23,22)(18,25,28,27)(19,30,21,31)(24,29,26,32);(9,11,12,15)(10,14,13,16)(17,21,23,19)(18,26,28,24)(20,31,22,30)(25,32,27,29)Polycyclic presentation:
2^9|a^2=efi,b^2=i,c^2=i,d^2=gi,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=ei,h^b=hi,e^d=ei,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 311472);