Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5956507 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1452 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,7,8,5)(10,11)(15,16)(17,18,21,24)(19,22,23,20)(25,26)(27,31)(28,36)(29,30)(32,37)(33,35)(34,39)(38,40);(3,8)(5,7)(9,10,14,11)(12,16,13,15)(19,23)(20,22)(26,30)(28,37)(31,33)(34,40);(9,11,14,10)(12,16,13,15)(17,19,21,23)(18,22,24,20)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40);(1,3)(2,5)(4,8)(6,7)(9,12,14,13)(10,15,11,16)(17,20,21,22)(18,23,24,19)(25,27,29,35)(26,31,30,33)(28,34,37,40)(32,38,36,39);(1,2)(3,5)(4,6)(7,8)(9,13,14,12)(10,16,11,15)(25,28,29,37)(26,32,30,36)(27,34,35,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=fh,d^2=gh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5905987);