Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4928752 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3129 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,11)(8,9)(10,12)(15,19)(18,20)(21,22)(23,28)(24,26)(25,27);(6,9)(10,11)(14,17)(18,20)(21,23,25,24)(22,26,27,28);(5,6,8,9)(7,10,12,11)(13,14)(15,18)(16,17)(19,20)(21,24,25,23)(22,26,27,28);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(21,22)(23,28)(24,26)(25,27);(5,7)(6,10)(8,12)(9,11)(13,15)(14,18)(16,19)(17,20)Polycyclic presentation:
2^9|b^2=fg,c^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,d^c=dg104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4835965);