Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13309 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #469 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #2034 |
| prime radical | C24 | C2×SD16 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
490 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,3,4)(5,6,12,16,13,17,9,8,10,15,30,29,19,27,11,28)(7,18,23,14,24,26,22,20,25,34,31,33,32,36,21,35);(5,7,19,24,10,25,13,32)(6,14,27,20,15,33,17,35)(8,26,16,34,28,36,29,18)(9,23,12,22,11,31,30,21);(5,8,9,27,19,16,12,15,10,28,11,17,13,29,30,6)(7,20,21,26,32,14,31,18,25,35,22,36,24,33,23,34)Polycyclic presentation:
2^9|a^2=g,b^2=eh,c^2=d,d^2=hi,f^2=hi,g^2=h,h^2=i,b^a=bd,c^a=ce,f^a=fi,c^b=cf,d^b=dh,e^b=ei,f^b=fh,g^b=gh,h^b=hi,f^c=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72321518, #72321571, #72321603, … |
| maximal normal subgroup of | #72321518, #72321571, #72321603, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56265);