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 | |
1785 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,6)(3,4,8,7)(9,10,13,15)(11,12,16,14)(17,18,22,23)(19,24,21,20)(25,26)(27,32)(28,34)(29,31)(30,33)(35,39)(36,40)(37,38);(2,6)(3,8)(9,11)(10,14)(12,15)(13,16)(17,19)(18,20)(21,22)(23,24)(25,27,31,28)(26,32,29,34)(30,36,38,35)(33,40,37,39);(3,8)(4,7)(11,16)(12,14)(17,20,22,24)(18,19,23,21)(25,28,31,27)(26,32,29,34)(30,36,38,35)(33,39,37,40);(1,3)(2,7)(4,6)(5,8)(9,12)(10,11)(13,14)(15,16)(17,21,22,19)(18,20,23,24)(25,29,31,26)(27,34,28,32)(30,37,38,33)(35,40,36,39);(1,4,5,7)(2,3,6,8)(9,11)(10,12)(13,16)(14,15)(17,20)(18,19)(21,23)(22,24)(25,30,31,38)(26,33,29,37)(27,35,28,36)(32,39,34,40)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fi,d^2=fi,e^2=fgh,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, 5234104);