Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| automorphism | Aut(G) order 2097152 |
| 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 | C26 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1984 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,7)(4,8)(5,6)(9,10)(11,16)(12,13)(14,15)(19,23)(22,24)(25,26,30,33)(27,28,34,31)(29,37,38,32)(35,39,40,36);(2,6)(3,8)(9,11,14,13)(10,12,15,16)(18,21)(22,24)(27,34)(28,31)(35,40)(36,39);(2,6)(4,7)(10,15)(11,13)(17,18,20,21)(19,22,23,24)(25,27)(26,31)(28,33)(29,35)(30,34)(32,39)(36,37)(38,40);(1,3)(2,7)(4,6)(5,8)(9,12)(10,11)(13,15)(14,16)(25,28,30,31)(26,27,33,34)(29,36,38,39)(32,35,37,40);(1,4,5,7)(2,3,6,8)(9,13,14,11)(10,12,15,16)(17,19)(18,22)(20,23)(21,24)(25,29,30,38)(26,32,33,37)(27,35,34,40)(28,36,31,39)Polycyclic presentation:
2^9|a^2=fg,b^2=fgi,c^2=hi,d^2=fg,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4557672);