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 | |
1738 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)(17,18,21,24)(19,22,23,20)(26,28)(27,33)(29,37)(38,40);(11,16)(13,15)(17,19)(18,22)(20,24)(21,23)(25,26,30,28)(27,31,33,34)(29,36,37,32)(35,40,39,38);(2,3)(6,7)(17,20,21,22)(18,23,24,19)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39);(1,3,5,2)(4,7,8,6)(9,11)(10,13)(12,16)(14,15)(17,20)(18,19)(21,22)(23,24)(25,28,30,26)(27,34,33,31)(29,36,37,32)(35,40,39,38);(1,4)(2,6)(3,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,20)(21,23)(22,24)(25,29,30,37)(26,32,28,36)(27,35,33,39)(31,38,34,40)Polycyclic presentation:
2^9|a^2=gi,b^2=f,c^2=h,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, 5905957);