Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D4×D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×C4×D8 |
| prime radical | C22×#508 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C22⋊C8 ◁ C8×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
979 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,9,14,11,15,10,26)(7,16,19,32,20,33,27,17)(8,21,23,31,25,35,24,13)(12,22,28,34,30,36,29,18);(5,7)(6,12)(8,22)(9,19)(10,27)(11,20)(13,17)(14,28)(15,30)(16,21)(18,24)(23,34)(25,36)(26,29)(31,32)(33,35);(2,4);(5,6)(7,17)(8,21)(9,14)(10,26)(11,15)(12,22)(13,24)(16,19)(18,29)(20,32)(23,31)(25,35)(27,33)(28,34)(30,36)Polycyclic presentation:
2^9|a^2=g,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eh,h^a=hi,d^b=de,e^b=eh,h^b=hi,e^d=eh,h^d=hi110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404646);