Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #180864 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#376 |
| prime radical | C25 | #41 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1782 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)(7,16,19,30)(8,13)(9,14)(10,24)(11,15)(12,18,26,20)(17,29,31,36)(21,27)(22,28)(23,34)(25,32,35,33);(5,7,10,20,11,19,9,18)(6,12,14,16,15,26,24,30)(8,17,23,33,21,31,22,32)(13,25,28,29,27,35,34,36);(2,4)(5,8,9,22,11,21,10,23)(6,13,14,28,15,27,24,34)(7,17,18,32,19,31,20,33)(12,25,16,29,26,35,30,36);(8,21)(13,27)(17,31)(22,23)(25,35)(28,34)(29,36)(32,33)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,h^a=hi,g^b=gi,d^c=di116 conjugacy classes — view the full character table.
| maximal subgroup of | #72252118, #72252124, #72252134, … |
| maximal normal subgroup of | #72252118, #72252124, #72252134, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260681);