Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×C4×C22⋊C4 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1236 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(9,10,14,18,15,19,13,28)(11,20,24,30,22,34,23,16)(12,17,27,32,25,31,26,38)(21,33,36,39,37,40,35,29);(1,2,3,6,4,5,7,8)(9,11)(10,16,19,30)(12,21)(13,23)(14,24)(15,22)(17,29,31,39)(18,20,28,34)(25,37)(26,35)(27,36)(32,33,38,40);(9,12,14,27,15,25,13,26)(10,17,18,32,19,31,28,38)(11,21,24,36,22,37,23,35)(16,29,20,33,30,39,34,40);(2,5)(6,8)(11,22)(12,25)(16,30)(17,31)(20,34)(23,24)(26,27)(32,38)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,e^2=h,f^2=h,g^2=i,b^a=be,e^a=eh,g^a=gh,d^b=dhi,d^c=dh128 conjugacy classes — view the full character table.
| maximal subgroup of | #72226709 |
| maximal normal subgroup of | #72226709 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 89511);