Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| 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 | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1788 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)(3,7,8,4)(9,10,13,15)(11,12,16,14)(17,18)(19,24)(20,23)(21,22)(25,26,30,28)(27,31,33,35)(29,36,38,32)(34,40,37,39);(1,3,5,8)(2,4,6,7)(9,11,13,16)(10,14,15,12)(18,21)(19,20)(26,28)(27,33)(32,36)(34,37);(2,6)(4,7)(10,15)(11,16)(17,19,22,20)(18,23,21,24)(25,27)(26,31)(28,35)(29,37)(30,33)(32,40)(34,38)(36,39);(1,3)(2,4)(5,8)(6,7)(9,12,13,14)(10,11,15,16)(17,20,22,19)(18,23,21,24)(25,28,30,26)(27,31,33,35)(29,36,38,32)(34,39,37,40);(1,4)(2,3)(5,7)(6,8)(9,11)(10,14)(12,15)(13,16)(17,21,22,18)(19,23,20,24)(25,29)(26,32)(27,34)(28,36)(30,38)(31,39)(33,37)(35,40)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,c^2=gi,d^2=h,e^2=gi,b^a=bf,c^a=cg,d^a=di,e^a=eh,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, 5111004);