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) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #14885 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2441 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(3,7)(9,10,12,13)(11,15,16,14)(17,18,20,23)(19,22,24,21)(25,26)(27,31)(28,30)(29,32)(33,35)(34,38)(36,37)(39,40);(2,5)(6,8)(10,13)(14,15)(17,19,20,24)(18,21,23,22)(26,28)(27,33)(29,37)(38,40);(9,11)(10,14)(12,16)(13,15)(17,18)(19,22)(20,23)(21,24)(25,27,30,33)(26,31,28,35)(29,34,37,39)(32,38,36,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,16)(13,15)(17,19)(18,22)(20,24)(21,23)(25,28,30,26)(27,31,33,35)(29,36,37,32)(34,38,39,40);(1,3)(2,6)(4,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,23)(21,24)(25,29,30,37)(26,32,28,36)(27,34,33,39)(31,38,35,40)Polycyclic presentation:
2^9|a^2=g,b^2=fgh,c^2=fg,d^2=fg,e^2=fg,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, 5866657);