Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
1820 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)(4,8)(5,6)(9,10)(11,12)(13,15)(14,16)(17,18,21,23)(19,20,24,22)(26,30)(31,33)(32,37)(38,39);(2,6)(3,8)(9,11,13,16)(10,14,15,12)(17,19)(18,22)(20,23)(21,24)(28,36)(32,37)(34,40)(38,39);(3,8)(4,7)(11,16)(12,14)(19,24)(20,22)(25,26,29,30)(27,33,35,31)(28,32,36,37)(34,39,40,38);(1,3)(2,4)(5,8)(6,7)(9,11)(10,12)(13,16)(14,15)(17,20)(18,19)(21,22)(23,24)(25,27,29,35)(26,31,30,33)(28,34,36,40)(32,38,37,39);(1,4)(2,3)(5,7)(6,8)(9,12)(10,11)(13,14)(15,16)(17,19)(18,20)(21,24)(22,23)(25,28)(26,32)(27,34)(29,36)(30,37)(31,38)(33,39)(35,40)Polycyclic presentation:
2^9|a^2=fhi,b^2=fi,c^2=g,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5143710);