Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30396 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1564 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(5,6,8,7)(9,10)(11,15)(12,14)(13,16)(17,18,21,25,22,26,19,23)(20,28,27,31,29,32,30,24);(1,2)(3,4)(9,11)(10,13)(12,16)(14,15)(17,19,22,21)(18,23,26,25)(20,27,29,30)(24,28,31,32);(2,4)(5,7,8,6)(18,23)(19,21)(24,28)(25,26)(27,30)(31,32);(10,14)(11,16)(17,20,22,29)(18,24,26,31)(19,27,21,30)(23,28,25,32)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=fhi,d^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,f^c=fi,h^c=hi,f^d=fi,h^d=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 296205);