Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5956507 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1994 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10,12,14)(11,15,16,13)(17,18)(19,21)(20,23)(22,24)(25,26,30,27)(28,33,35,31)(29,32,37,34)(36,39,40,38);(3,8)(5,7)(17,19)(18,21)(20,24)(22,23)(26,27)(28,35)(29,37)(38,39);(9,11,12,16)(10,13,14,15)(17,18,20,23)(19,22,24,21)(25,27,30,26)(28,31,35,33)(29,34,37,32)(36,38,40,39);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,16)(14,15)(17,18)(19,22)(20,23)(21,24)(25,28,30,35)(26,31,27,33)(29,36,37,40)(32,38,34,39);(1,2)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,22)(20,24)(21,23)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40)Polycyclic presentation:
2^9|a^2=i,c^2=fg,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5894892);