Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2031997 |
| 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 | |
1508 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)(3,7,8,4)(17,18,20,21)(19,22,23,24)(25,26,30,32)(27,33,35,29)(28,31,38,40)(34,39,36,37);(2,6)(3,8)(10,13)(14,16)(18,21)(19,23)(25,27,30,35)(26,29,32,33)(28,36,38,34)(31,39,40,37);(1,3,5,8)(2,7,6,4)(9,10,12,13)(11,14,15,16)(17,19)(18,22)(20,23)(21,24)(25,27)(26,29)(28,34)(30,35)(31,37)(32,33)(36,38)(39,40);(1,4,5,7)(2,8,6,3)(25,28,30,38)(26,31,32,40)(27,34,35,36)(29,37,33,39);(1,2)(3,7)(4,8)(5,6)(9,11,12,15)(10,14,13,16)(17,18)(19,22)(20,21)(23,24)(25,29)(26,27)(28,37)(30,33)(31,34)(32,35)(36,40)(38,39)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fhi,d^2=gi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1979073);