Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15273 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2371 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(6,8)(10,13)(11,15)(18,19)(20,24)(25,26)(27,28)(29,30)(31,32);(10,13)(14,16)(17,18,21,19)(20,22,24,23)(26,30)(28,32);(9,10)(11,14)(12,13)(15,16)(17,18,21,19)(20,23,24,22)(25,27,29,31)(26,28,30,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,21,18)(20,22,24,23)(25,28,29,32)(26,27,30,31);(1,3,4,7)(2,6,5,8)(9,11,12,15)(10,14,13,16)(17,20)(18,22)(19,23)(21,24)(25,27)(26,28)(29,31)(30,32)Polycyclic presentation:
2^9|b^2=gh,c^2=f,d^2=f,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3522101);