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 | |
1506 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,6,8)(3,4,7,5)(10,13)(11,15)(25,26,30,31)(27,32,35,38)(28,36,37,33)(29,34,39,40);(1,3,6,7)(2,4,8,5)(10,13)(14,16)(17,18,20,22)(19,21,23,24)(26,31)(27,35)(33,36)(34,40);(1,4,6,5)(2,3,8,7)(9,10)(11,14)(12,13)(15,16)(17,18)(19,21)(20,22)(23,24)(25,27)(26,32)(28,34)(29,33)(30,35)(31,38)(36,39)(37,40);(1,5,6,4)(2,7,8,3)(25,28,30,37)(26,33,31,36)(27,34,35,40)(29,38,39,32);(1,2)(3,4)(5,7)(6,8)(9,11,12,15)(10,14,13,16)(17,19,20,23)(18,21,22,24)(25,29)(26,34)(27,33)(28,32)(30,39)(31,40)(35,36)(37,38)Polycyclic presentation:
2^9|a^2=i,b^2=fgh,c^2=fi,d^2=i,e^2=fh,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, 1233579);