Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #140370 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1943 |
| 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 | |
478 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,12,16,13,19,9,25)(7,20,23,8,24,26,21,14)(10,30,29,18,11,32,28,17)(15,31,36,27,33,34,22,35);(5,7,9,21,13,24,12,23)(6,14,16,20,19,8,25,26)(10,15,28,22,11,33,29,36)(17,34,30,35,18,31,32,27);(2,4)(5,8,9,26,13,14,12,20)(6,15,16,22,19,33,25,36)(7,17,21,30,24,18,23,32)(10,31,28,27,11,34,29,35)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=d,d^2=i,e^2=h,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,f^a=fh,h^a=hi,c^b=cf,e^b=eh,f^b=fhi,h^b=hi,e^c=eh,f^c=fhi,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250495, #72250516, #72250519, … |
| maximal normal subgroup of | #72250495, #72250516, #72250519, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55966);