Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13490 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2073 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | #135 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
438 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)(6,12)(7,17)(8,21)(9,24)(11,23)(14,26)(15,22)(16,30)(18,32)(19,34)(20,31)(25,27)(28,35)(33,36);(5,6,8,22,23,26,9,14,10,15,24,12,11,16,21,30)(7,13,18,33,32,35,17,27,20,29,34,25,19,28,31,36);(5,7,8,18,23,32,9,17,10,20,24,34,11,19,21,31)(6,13,12,25,26,35,30,36,15,29,22,33,16,28,14,27)Polycyclic presentation:
2^9|a^2=g,b^2=eh,c^2=d,d^2=hi,e^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,h^a=hi,c^b=cf,d^b=di,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72322143, #72322145, #72322163, … |
| maximal normal subgroup of | #72322143, #72322145, #72322163, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56206);