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 | |
1576 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)(9,10,13,16)(11,14,12,15)(19,23)(22,24)(25,26,30,32)(27,33,35,28)(29,31,37,38)(34,39,40,36);(1,3,5,8)(2,4,6,7)(9,11,13,12)(10,14,16,15)(18,21)(22,24)(29,37)(31,38)(34,40)(36,39);(2,6)(4,7)(9,11,13,12)(10,14,16,15)(17,18)(19,22)(20,21)(23,24)(25,27,30,35)(26,28,32,33)(29,34,37,40)(31,36,38,39);(1,3)(2,4)(5,8)(6,7)(9,10)(11,14)(12,15)(13,16)(25,28)(26,27)(29,36)(30,33)(31,34)(32,35)(37,39)(38,40);(1,4)(2,3)(5,7)(6,8)(9,12,13,11)(10,15,16,14)(17,19)(18,22)(20,23)(21,24)(25,29,30,37)(26,31,32,38)(27,34,35,40)(28,36,33,39)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,c^2=gi,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, 5108268);