Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13432 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2044 |
| commutator | C2×#41 | C22×C4 |
| Frattini, prime radical, cosocle | C22×#41 | C23 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
663 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)(6,11)(7,15,17,30)(8,18)(12,25,27,36)(13,23)(14,24)(16,31,32,29)(19,28)(20,33)(22,26,35,34);(5,6,8,11,10,14,18,24)(7,12,16,22,17,27,32,35)(9,20,19,33,21,13,28,23)(15,26,29,25,30,34,31,36);(5,7,8,16,10,17,18,32)(6,12,11,22,14,27,24,35)(9,15,19,29,21,30,28,31)(13,26,23,25,20,34,33,36)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=d,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fhi,c^b=cf,e^b=eh,g^c=gi,h^c=hi,f^e=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72311939, #72311949, #72311967, … |
| maximal normal subgroup of | #72311939, #72311949, #72311967, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56020);