Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32935 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C22×C4⋊C4 |
| prime radical | C24 | C22×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
866 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,21,24,22,25,20,30)(19,26,29,31,27,32,28,23);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(19,27)(23,31)(26,32)(28,29);(17,19,20,28,22,27,21,29)(18,23,24,26,25,31,30,32);(1,4,5,3)(2,6,7,8)(10,14)(13,16)Polycyclic presentation:
2^9|a^2=h,b^2=eg,c^2=f,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,c^b=ci104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299719);