Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4932388 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2620 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)(7,12)(8,11)(9,10)(13,14,18,17)(15,20,19,16)(22,25)(26,27);(6,9)(11,12)(14,17)(15,19);(5,7,10,8)(6,11,9,12)(13,15)(14,16)(17,20)(18,19)(21,22)(23,27)(24,25)(26,28);(1,2)(3,4)(5,8,10,7)(6,12,9,11)(13,16,18,20)(14,19,17,15)(21,23,24,28)(22,26,25,27);(5,9,10,6)(7,12,8,11)(13,17,18,14)(15,20,19,16)Polycyclic presentation:
2^9|a^2=fhi,c^2=fh,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4849206);