Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3128 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)(3,4)(6,7)(9,10,14,12)(11,19,22,16)(13,17,27,24)(15,18,29,25)(20,23,35,30)(21,37,39,32)(26,38,40,33)(28,34,31,36);(5,6,8,7)(9,11,14,22)(10,16,12,19)(13,26,27,40)(15,30,29,23)(17,33,24,38)(18,35,25,20)(21,31,39,28)(32,36,37,34);(2,4)(5,7,8,6)(9,12,14,10)(11,20,22,35)(13,24,27,17)(15,18,29,25)(16,30,19,23)(21,38,39,33)(26,37,40,32)(28,34,31,36);(9,13,14,27)(10,17,12,24)(11,21,22,39)(15,31,29,28)(16,32,19,37)(18,36,25,34)(20,38,35,33)(23,40,30,26);(10,12)(11,22)(17,24)(18,25)(21,39)(23,30)(26,40)(34,36)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=f,d^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345919);