Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2021 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C24 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C23 | #385 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1658 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)(6,8)(9,10,13,18,14,17,25,32)(11,19,22,33,21,15,26,28)(12,16,24,30,23,31,36,40)(20,27,35,38,34,29,37,39);(2,4)(9,11)(10,15,17,19)(12,20,23,34)(13,22)(14,21)(16,29)(18,28,32,33)(24,35,36,37)(25,26)(27,31)(30,39)(38,40);(5,6,7,8)(9,12)(10,16)(11,20,21,34)(13,24)(14,23)(15,27,19,29)(17,31)(18,30)(22,35,26,37)(25,36)(28,38,33,39)(32,40)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,g^2=i,b^a=bd,c^a=ce,f^a=fi,h^a=hi,c^b=cf,d^b=di,e^b=ei,f^b=fi,f^c=fi,h^c=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72314501, #72314514, #72314526, … |
| maximal normal subgroup of | #72314501, #72314514, #72314526, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31236);