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) = C25 |
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 | #33600 | 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 | |
627 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)(5,6,10,16,11,17,9,15,12,18,27,34,20,8,14,28)(7,19,23,13,24,31,29,26,25,21,33,32,30,36,22,35);(5,7,20,24,12,25,11,30)(6,13,8,26,18,32,17,35)(9,23,10,29,14,33,27,22)(15,31,16,21,28,36,34,19);(2,4);(5,8,27,15,11,6,14,34,12,17,10,28,20,18,9,16)(7,21,23,32,24,36,29,35,25,19,33,13,30,31,22,26)Polycyclic presentation:
2^9|a^2=g,b^2=egh,d^2=eh,e^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^b=de,e^b=eh,g^b=gh,h^b=hi110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405539);