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 | |
1588 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)(3,7)(4,8)(5,6)(9,10)(11,12)(13,14)(15,16)(17,18,21,24)(19,22,20,23)(25,26,30,27)(28,33,36,31)(29,32,37,34)(35,39,40,38);(1,3,5,8)(2,4,6,7)(9,10,13,14)(11,16,15,12)(26,27)(29,37)(31,33)(35,40);(2,6)(4,7)(10,14)(11,15)(17,19,21,20)(18,22,24,23)(25,27,30,26)(28,31,36,33)(29,34,37,32)(35,38,40,39);(1,4)(2,3)(5,7)(6,8)(9,11,13,15)(10,12,14,16)(17,18)(19,22)(20,23)(21,24)(25,28,30,36)(26,31,27,33)(29,35,37,40)(32,38,34,39);(1,3)(2,4)(5,8)(6,7)(9,12)(10,15)(11,14)(13,16)(17,20,21,19)(18,23,24,22)(25,29)(26,32)(27,34)(28,35)(30,37)(31,38)(33,39)(36,40)Polycyclic presentation:
2^9|a^2=fhi,b^2=g,c^2=fhi,d^2=h,e^2=ghi,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, 1885255);