Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| 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 | |
1493 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,4)(3,7,8,6)(11,15)(12,16)(25,26,30,29)(27,31,33,35)(28,36,38,32)(34,40,37,39);(9,10,13,14)(11,16,15,12)(18,22)(23,24)(26,29)(27,33)(32,36)(34,37);(1,2,5,4)(3,6,8,7)(9,11)(10,12)(13,15)(14,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,31)(28,37)(29,35)(30,33)(32,40)(34,38)(36,39);(1,3,5,8)(2,6,4,7)(9,10)(11,12)(13,14)(15,16)(17,19,21,20)(18,23,22,24)(25,28)(26,32)(27,34)(29,36)(30,38)(31,39)(33,37)(35,40);(1,4,5,2)(3,7,8,6)(9,12)(10,11)(13,16)(14,15)(17,20,21,19)(18,24,22,23)(25,29,30,26)(27,35,33,31)(28,36,38,32)(34,40,37,39)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=fi,d^2=ghi,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1687784);