Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13472 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2067 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | #137 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
484 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)(5,6,11,17,9,16,26,15,13,18,27,28,12,8,10,25)(7,19,23,34,20,14,29,31,24,33,36,30,22,32,21,35);(5,7,9,20,13,24,12,22)(6,14,8,19,18,32,16,33)(10,21,11,23,26,29,27,36)(15,30,17,31,25,34,28,35);(2,4)(5,8,9,18,13,16,12,6)(7,14,20,19,24,32,22,33)(10,25,11,28,26,15,27,17)(21,31,23,34,29,35,36,30)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=d,d^2=i,e^2=h,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,c^b=cf,f^b=fi,e^c=eh,f^c=fi,g^c=gh,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72321477, #72321547, #72321585, … |
| maximal normal subgroup of | #72321477, #72321547, #72321585, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56053);