Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| automorphism | Aut(G) order 1048576 |
| 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 | |
1618 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)(4,5)(6,8)(9,10,13,14)(11,12,16,15)(17,18)(19,23)(20,22)(21,24)(25,26,30,29)(27,32,35,31)(28,33,36,37)(34,39,40,38);(3,8)(6,7)(9,11)(10,12)(13,16)(14,15)(17,18)(19,23)(20,22)(21,24)(25,27,30,35)(26,31,29,32)(28,34,36,40)(33,38,37,39);(2,5)(6,7)(10,14)(11,16)(17,18,20,22)(19,21,24,23)(25,27)(26,32)(28,34)(29,31)(30,35)(33,39)(36,40)(37,38);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,14)(13,16)(17,18)(19,21)(20,22)(23,24)(25,28)(26,33)(27,34)(29,37)(30,36)(31,38)(32,39)(35,40);(1,3)(2,6)(4,8)(5,7)(9,12,13,15)(10,11,14,16)(17,19,20,24)(18,21,22,23)(25,29,30,26)(27,32,35,31)(28,33,36,37)(34,38,40,39)Polycyclic presentation:
2^9|a^2=fi,b^2=f,c^2=fhi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4132768);