Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2254 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(5,6)(7,9)(8,11)(10,12)(13,14)(15,17)(16,19)(18,20);(6,7)(10,11)(13,15,18,19)(14,16,20,17)(21,22,24,26)(23,27,28,25);(5,6,9,7)(8,11,12,10)(13,16,18,17)(14,19,20,15)(21,23,24,28)(22,25,26,27);(5,7,9,6)(8,11,12,10)(13,17,18,16)(14,19,20,15);(5,8)(6,10)(7,11)(9,12)(13,14)(15,16)(17,19)(18,20)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=h,d^2=f,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dg,e^c=ef104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 483941);