Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13459 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #498 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #2084 |
| prime radical | C24 | #137 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
484 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)(5,6,11,16,13,17,12,30)(7,9,19,28,20,29,15,10)(8,21,26,18,23,33,27,32)(14,25,22,24,31,36,34,35);(5,7)(6,9)(8,22,23,34)(10,30)(11,19)(12,15)(13,20)(14,26,31,27)(16,28)(17,29)(18,25,32,36)(21,35,33,24);(2,4)(5,8,12,26,13,23,11,27)(6,14,16,34,17,31,30,22)(7,18,19,21,20,32,15,33)(9,25,10,24,29,36,28,35)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=dh,e^2=h,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fh,h^a=hi,c^b=cf,e^b=ehi,f^b=fhi,h^b=hi,d^c=di,e^c=eh,f^c=fhi,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250139, #72250141, #72250143, … |
| maximal normal subgroup of | #72250139, #72250141, #72250143, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55990);