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 | |
1702 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)(17,18,21,23)(19,24,20,22)(25,26)(27,33)(28,34)(29,32)(30,31)(35,39)(36,40)(37,38);(2,5)(6,8)(10,13)(11,16)(17,19,21,20)(18,22,23,24)(26,31)(27,34)(29,37)(36,39);(11,16)(14,15)(19,20)(22,24)(25,27)(26,28)(29,35)(30,34)(31,33)(32,36)(37,40)(38,39);(1,2,4,5)(3,6,7,8)(9,11)(10,14)(12,16)(13,15)(17,20,21,19)(18,22,23,24)(25,28)(26,27)(29,36)(30,33)(31,34)(32,35)(37,39)(38,40);(1,3)(2,6)(4,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,24)(20,22)(21,23)(25,29,30,37)(26,32,31,38)(27,35,34,40)(28,36,33,39)Polycyclic presentation:
2^9|a^2=fg,b^2=gh,d^2=fi,e^2=g,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, 5969492);