Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1258513 |
| 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 | |
1562 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,4)(6,7)(9,10,14,16)(11,12,15,13)(18,21)(19,23)(25,26)(27,32)(28,33)(29,34)(30,31)(35,37)(36,40)(38,39);(9,11,14,15)(10,12,16,13)(18,21)(22,24)(26,31)(27,33)(34,39)(35,40);(1,2,5,4)(3,6,8,7)(9,12,14,13)(10,11,16,15)(17,18,20,21)(19,22,23,24)(25,27,30,33)(26,32,31,28)(29,35,38,40)(34,37,39,36);(1,3,5,8)(2,6,4,7)(9,10)(11,12)(13,15)(14,16)(25,28,30,32)(26,33,31,27)(29,37,38,36)(34,35,39,40);(1,4,5,2)(3,7,8,6)(9,13,14,12)(10,15,16,11)(17,19,20,23)(18,22,21,24)(25,29)(26,34)(27,35)(28,36)(30,38)(31,39)(32,37)(33,40)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fg,d^2=ghi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1220337);