Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12952 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1864 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
950 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)(5,6,10,15,11,17,29,16)(7,18,21,28,22,35,12,32)(8,23,24,20,26,33,25,19)(9,27,30,13,31,34,36,14);(2,4)(5,7,10,21,11,22,29,12)(6,8,16,25,17,26,15,24)(9,23,30,20,31,33,36,19)(13,32,27,35,14,28,34,18);(5,8)(6,12)(7,16)(9,28)(10,25)(11,26)(13,20)(14,19)(15,22)(17,21)(18,30)(23,34)(24,29)(27,33)(31,32)(35,36)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=hi,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,g^a=gi,h^a=hi,d^b=dhi,e^b=eh,h^b=hi,d^c=dh,e^c=eh,f^c=fi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72250757, #72250764, #72250775, … |
| maximal normal subgroup of | #72250757, #72250764, #72250775, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53990);