Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1536 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,4)(5,6,8,7)(9,10,14,19,15,20,13,18)(11,21,25,16,23,32,24,31)(12,26,28,34,29,33,27,17)(22,36,39,40,37,30,38,35);(9,11,13,25,15,23,14,24)(10,16,18,32,20,31,19,21)(12,22,28,38,29,37,27,39)(17,30,26,40,34,36,33,35);(2,4)(9,12)(10,17)(11,22)(13,28)(14,27)(15,29)(16,30)(18,26)(19,33)(20,34)(21,35)(23,37)(24,39)(25,38)(31,36)(32,40);(5,7,8,6)(11,23)(16,31)(21,32)(22,37)(24,25)(30,36)(35,40)(38,39)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=eg,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^b=di,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261639);