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 | |
3140 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)(9,10,14,19)(11,20,24,16)(12,17,27,35)(13,28,15,18)(21,32,23,30)(22,37,39,31)(25,40,38,33)(26,36,29,34);(5,6,8,7)(9,11,14,24)(10,16,19,20)(12,25,27,38)(13,23,15,21)(17,33,35,40)(18,30,28,32)(22,29,39,26)(31,34,37,36);(2,4)(5,7,8,6)(11,21)(13,15)(16,30)(18,28)(20,32)(22,25)(23,24)(26,29)(31,33)(34,36)(37,40)(38,39);(9,12)(10,17)(11,22)(13,26)(14,27)(15,29)(16,31)(18,34)(19,35)(20,37)(21,25)(23,38)(24,39)(28,36)(30,33)(32,40);(9,13,14,15)(10,18,19,28)(11,23,24,21)(12,26,27,29)(16,30,20,32)(17,34,35,36)(22,38,39,25)(31,33,37,40)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,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, 6345899);