Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2735949 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1870 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,6)(3,7)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(26,27)(28,35)(29,37)(39,40);(1,2)(3,8)(4,7)(5,6)(10,14)(11,16)(18,21)(22,23)(25,26,30,27)(28,31,35,33)(29,32,37,34)(36,39,38,40);(2,6)(3,7)(9,11,13,16)(10,15,14,12)(17,19)(18,22)(20,24)(21,23)(25,27,30,26)(28,31,35,33)(29,34,37,32)(36,39,38,40);(1,3)(2,4)(5,7)(6,8)(9,11)(10,15)(12,14)(13,16)(17,18)(19,23)(20,21)(22,24)(25,28)(26,31)(27,33)(29,38)(30,35)(32,40)(34,39)(36,37);(1,4)(2,7)(3,6)(5,8)(9,12,13,15)(10,16,14,11)(17,19)(18,22)(20,24)(21,23)(25,29)(26,32)(27,34)(28,36)(30,37)(31,39)(33,40)(35,38)Polycyclic presentation:
2^9|b^2=g,c^2=hi,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2666337);