Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #188759 |
| 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 | |
586 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,10,17,9,16,26,33,11,14,28,13,27,15,8,24)(7,18,22,29,21,30,36,32,23,34,19,35,20,12,25,31);(5,7,10,22,9,21,26,36,11,23,28,19,27,20,8,25)(6,12,17,31,16,18,33,29,14,30,13,32,15,34,24,35);(2,4)(5,8,9,10,11,26,27,28)(6,13,15,33,14,17,16,24)(7,19,20,36,23,22,21,25)(12,29,18,32,30,35,34,31);(6,14)(12,30)(13,17)(15,16)(18,34)(24,33)(29,35)(31,32)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=e,e^2=i,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,c^b=ch,e^b=ei,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72239414, #72239422, #72239428, … |
| maximal normal subgroup of | #72239414, #72239422, #72239428, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401973);