Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C26×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C22 | C23×C42 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3566 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(5,6,8,7)(9,10)(11,15)(12,14)(13,16)(19,23)(22,24);(1,2,4,3)(5,6,8,7)(9,10,12,14)(11,13,16,15)(17,18,21,20)(19,22,23,24);(9,11)(10,13)(12,16)(14,15)(17,19,21,23)(18,22,20,24);(9,10)(11,13)(12,14)(15,16)(17,18,21,20)(19,22,23,24);(1,3,4,2)(5,7,8,6)(9,11)(10,13)(12,16)(14,15)(17,20,21,18)(19,24,23,22)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=fg,d^2=fg,e^2=fi,b^a=bf,c^a=cg,e^a=ef,d^b=df176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420929);