Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2031997 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1494 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,6)(3,4,8,7)(9,10,14,15)(11,13,12,16)(19,23)(21,24)(25,26)(27,30)(28,33)(29,32)(31,35)(34,37)(36,39)(38,40);(2,6)(4,7)(10,15)(13,16)(17,18,20,22)(19,21,23,24)(26,27)(28,35)(29,37)(38,39);(9,11,14,12)(10,13,15,16)(17,18)(19,21)(20,22)(23,24)(25,27,30,26)(28,33,35,31)(29,34,37,32)(36,39,40,38);(1,3)(2,7)(4,6)(5,8)(9,12,14,11)(10,16,15,13)(17,19)(18,21)(20,23)(22,24)(25,28)(26,31)(27,33)(29,36)(30,35)(32,38)(34,39)(37,40);(1,4,5,7)(2,3,6,8)(9,13,14,16)(10,11,15,12)(25,29,30,37)(26,32,27,34)(28,36,35,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=fg,b^2=gh,c^2=fhi,d^2=fghi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1978606);