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 | |
1699 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)(9,10,13,12)(11,15,16,14)(17,18,20,22)(19,21,23,24)(26,30)(27,33)(28,36)(38,40);(1,3)(2,6)(4,7)(5,8)(10,12)(14,15)(19,23)(21,24)(25,26,29,30)(27,34,33,31)(28,37,36,32)(35,38,39,40);(3,8)(6,7)(9,10,13,12)(11,15,16,14)(17,19)(18,21)(20,23)(22,24)(25,26)(27,34)(28,32)(29,30)(31,33)(35,40)(36,37)(38,39);(1,3)(2,7)(4,6)(5,8)(9,11,13,16)(10,14,12,15)(17,18)(19,21)(20,22)(23,24)(25,27,29,33)(26,31,30,34)(28,35,36,39)(32,38,37,40);(1,4,5,2)(3,6,8,7)(9,12,13,10)(11,15,16,14)(25,28,29,36)(26,32,30,37)(27,35,33,39)(31,38,34,40)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=fghi,d^2=fhi,e^2=f,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, 1625271);