Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12711 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1781 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×#41 | C23 |
| socle | C2 | #10761 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#568 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1030 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,13,16,28)(4,18,21,10)(5,22,23,9)(6,11)(7,12)(8,14)(15,31)(17,29,32,24)(19,26)(20,27)(25,30);(1,3,6,17,7,16,20,32)(2,8,11,25,12,26,27,31)(4,19,5,15,21,14,23,30)(9,24,18,13,22,29,10,28);(2,9)(8,24)(10,27)(11,18)(12,22)(13,25)(26,29)(28,31)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,h^a=hi,d^b=di,f^b=fhi,f^c=fh,f^d=fi,f^e=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72317336, #72317364, #72317395, … |
| maximal normal subgroup of | #72317336, #72317364, #72317395, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53893);