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) = C25×C4 |
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 | |
1738 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,4)(3,6,7,8)(9,10,14,15)(11,16,13,12)(17,18)(19,23)(20,21)(22,24)(26,30)(28,36)(31,33)(34,40);(2,4)(3,7)(9,11)(10,12)(13,14)(15,16)(17,19,21,24)(18,22,20,23)(26,30)(28,36)(31,33)(34,40);(2,4)(6,8)(10,15)(11,13)(17,18,21,20)(19,22,24,23)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40);(1,3,5,7)(2,6,4,8)(9,12,14,16)(10,11,15,13)(17,20,21,18)(19,22,24,23)(25,27,29,35)(26,31,30,33)(28,34,36,40)(32,38,37,39);(1,4,5,2)(3,8,7,6)(9,13,14,11)(10,16,15,12)(17,19)(18,23)(20,22)(21,24)(25,28,29,36)(26,32,30,37)(27,34,35,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=i,d^2=g,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, 5108364);