Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C23 | #386 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1630 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,4)(5,6,7,8)(9,10)(11,21)(12,26)(13,19)(14,18)(15,20)(16,23)(17,28)(22,37)(24,33)(25,32)(27,40)(29,35)(30,36)(31,39)(34,38);(2,4)(9,11,14,24,15,25,13,23)(10,16,19,32,20,33,18,21)(12,27,28,22,30,38,29,39)(17,34,26,37,35,40,36,31);(9,12,13,28,15,30,14,29)(10,17,18,26,20,35,19,36)(11,22,23,38,25,39,24,27)(16,31,21,34,33,37,32,40)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,d^2=i,e^2=i,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,h^a=hi,c^b=cf,f^c=fi,h^c=hi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31168);