Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4311730 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1428 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(6,8)(11,15)(14,16)(19,20)(23,24)(25,26,29,30)(27,31,35,33)(28,36,37,32)(34,39,40,38);(1,2,5,4)(3,6,7,8)(9,10,13,12)(11,14,15,16)(18,22)(23,24)(26,30)(28,37)(31,33)(34,40);(1,3)(2,6)(4,8)(5,7)(9,11,13,15)(10,14,12,16)(17,18,21,22)(19,23,20,24)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40);(1,3)(2,6)(4,8)(5,7)(17,19,21,20)(18,23,22,24)(25,27,29,35)(26,31,30,33)(28,34,37,40)(32,38,36,39);(1,4,5,2)(3,8,7,6)(9,12,13,10)(11,16,15,14)(17,20,21,19)(18,24,22,23)(25,28,29,37)(26,32,30,36)(27,34,35,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=fgi,d^2=h,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4240837);