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 | |
478 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)(5,6,11,15,12,18,9,24)(7,19,22,8,23,25,20,13)(10,29,32,17,33,35,28,16)(14,31,34,26,30,36,21,27);(5,7,9,20,12,23,11,22)(6,13,15,19,18,8,24,25)(10,30,28,34,33,14,32,21)(16,31,29,26,17,36,35,27);(2,4)(5,8,9,25,12,13,11,19)(6,14,15,21,18,30,24,34)(7,16,20,29,23,17,22,35)(10,31,28,26,33,36,32,27)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=fhi,h^a=hi,c^b=cf,e^b=ehi,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 | #72321294, #72321352, #72321379, … |
| maximal normal subgroup of | #72321294, #72321352, #72321379, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55974);