Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #188761 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#117 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×#432 |
| prime radical | C24 | C2×M16 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
596 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10,12,18,13,19,26,35,14,17,25,33,27,16,30,32)(11,20,22,34,23,31,37,40,21,15,24,29,36,28,38,39);(1,3,4,7)(2,5,6,8)(9,11)(10,15)(12,24)(13,23)(14,21)(16,31)(17,20)(18,34)(19,28)(22,25)(26,38)(27,36)(29,33)(30,37)(32,39)(35,40);(2,6)(5,8);(10,16,17,19)(11,21)(12,25)(15,28,20,31)(18,35,33,32)(23,36)(26,30)(29,40,34,39)Polycyclic presentation:
2^9|a^2=g,b^2=f,d^2=e,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,d^b=di,g^b=gi,g^d=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72237213, #72239207, #72239473, … |
| maximal normal subgroup of | #72237213, #72239207, #72239473, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401320);