Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476293 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
787 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,21)(12,23)(13,28,32,39)(14,34)(15,20)(16,19)(17,26)(18,33,38,31)(22,40,37,25)(24,27)(29,36,35,30);(1,3,4,7)(2,5,6,8)(9,11,15,24)(10,12,20,26)(13,29,32,35)(14,17,16,23)(18,36,38,30)(19,27,34,21)(22,31,37,33)(25,28,40,39);(2,6)(5,8)(9,12,15,26)(10,17,20,23)(11,19,24,34)(13,30,32,36)(14,27,16,21)(18,37,38,22)(25,33,40,31)(28,35,39,29);(3,7)(5,8)(9,13)(10,18)(11,22)(12,25)(14,33)(15,32)(16,31)(17,35)(19,39)(20,38)(21,36)(23,29)(24,37)(26,40)(27,30)(28,34)Polycyclic presentation:
2^9|a^2=efg,b^2=f,c^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=def,e^b=ei,g^b=gi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei,h^d=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 385603);