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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1452 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,6)(3,4,8,7)(11,15)(14,16)(25,26)(27,32)(28,29)(30,31)(33,38)(34,39)(35,40)(36,37);(1,2)(3,7)(4,8)(5,6)(10,13)(14,16)(17,18,21,20)(19,23,24,22)(26,31)(27,34)(29,33)(35,37);(2,6)(3,8)(9,10,12,13)(11,14,15,16)(17,19,21,24)(18,22,20,23)(25,27,30,34)(26,32,31,39)(28,37,38,35)(29,36,33,40);(1,3)(2,4)(5,8)(6,7)(9,11,12,15)(10,14,13,16)(17,20,21,18)(19,23,24,22)(25,28)(26,33)(27,35)(29,31)(30,38)(32,36)(34,37)(39,40);(1,4)(2,3)(5,7)(6,8)(25,29,30,33)(26,28,31,38)(27,36,34,40)(32,37,39,35)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=gh,d^2=fh,e^2=fg,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, 1482805);