Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1718 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)(9,10,14,15)(11,13,12,16)(17,18)(19,20)(21,23)(22,24)(27,33)(28,35)(31,36)(32,37);(2,6)(3,8)(9,11,14,12)(10,13,15,16)(17,19)(18,23)(20,24)(21,22)(26,30)(31,36)(32,37)(38,40);(3,8)(4,7)(10,15)(11,12)(18,24)(19,21)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(1,3,5,8)(2,7,6,4)(9,12,14,11)(10,13,15,16)(17,20)(18,19)(21,24)(22,23)(25,27,29,33)(26,31,30,36)(28,34,35,39)(32,38,37,40);(1,4,5,7)(2,8,6,3)(9,13)(10,11)(12,15)(14,16)(17,21,22,19)(18,20,24,23)(25,28)(26,32)(27,34)(29,35)(30,37)(31,38)(33,39)(36,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=ghi,d^2=fh,e^2=f,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, 4976445);