Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13309 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #469 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #2076 |
| prime radical | C25 | Q16 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
712 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,9,15,11,14,10,25)(7,16,20,32,19,30,21,35)(8,22,23,29,24,36,17,13)(12,26,28,34,27,33,31,18);(1,2)(3,4)(5,7,17,33,10,21,24,34,11,19,23,26,9,20,8,18)(6,12,22,32,15,28,29,30,14,27,36,35,25,31,13,16);(2,4)(12,27)(16,30)(28,31)(32,35)Polycyclic presentation:
2^9|a^2=g,b^2=dh,d^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,e^b=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72257244, #72257250, #72257252, … |
| maximal normal subgroup of | #72257244, #72257250, #72257252, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55516);