Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #195990 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator, prime radical | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1082 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)(3,7)(4,6)(5,8)(17,18,20,24,21,23,25,31)(19,22,27,29,26,30,28,32);(1,3,4,8)(2,5,6,7)(9,10)(11,14)(12,13)(15,16);(10,13)(14,16)(17,19,25,27,21,26,20,28)(18,22,24,32,23,30,31,29);(2,6)(5,7)(9,11,12,15)(10,14,13,16)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=fhi,d^2=g,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,f^c=fi,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 282847);