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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1796 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,6)(3,7,8,4)(9,10,13,15)(11,12,16,14)(18,20)(19,21)(26,30)(28,35)(31,36)(34,39);(1,3,5,8)(2,4,6,7)(9,11)(10,14)(12,15)(13,16)(17,18)(19,23)(20,22)(21,24)(27,33)(31,36)(34,39)(38,40);(2,6)(4,7)(10,15)(11,16)(17,19,22,21)(18,23,20,24)(25,26,29,30)(27,31,33,36)(28,32,35,37)(34,38,39,40);(1,4)(2,3)(5,7)(6,8)(9,10)(11,12)(13,15)(14,16)(17,20,22,18)(19,23,21,24)(25,27,29,33)(26,31,30,36)(28,34,35,39)(32,38,37,40);(1,3)(2,4)(5,8)(6,7)(9,12)(10,11)(13,14)(15,16)(17,21,22,19)(18,23,20,24)(25,28)(26,32)(27,34)(29,35)(30,37)(31,38)(33,39)(36,40)Polycyclic presentation:
2^9|a^2=fhi,b^2=fi,c^2=fghi,d^2=fghi,e^2=hi,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, 1914613);