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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1706 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)(3,5)(4,7)(6,8)(9,10,12,14)(11,15,16,13)(17,18,22,20)(19,21,23,24)(25,26,30,27)(28,31,36,33)(29,34,37,32)(35,39,40,38);(11,16)(13,15)(17,19)(18,21)(20,24)(22,23)(26,27)(29,37)(31,33)(35,40);(2,3)(6,7)(18,20)(19,23)(25,27,30,26)(28,31,36,33)(29,34,37,32)(35,38,40,39);(1,3,5,2)(4,7,8,6)(9,11)(10,13)(12,16)(14,15)(17,20,22,18)(19,21,23,24)(25,28)(26,31)(27,33)(29,35)(30,36)(32,38)(34,39)(37,40);(1,4,5,8)(2,6,3,7)(9,10)(11,13)(12,14)(15,16)(17,21)(18,19)(20,23)(22,24)(25,29,30,37)(26,32,27,34)(28,35,36,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=fgi,c^2=f,d^2=g,e^2=gh,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, 5910005);