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 | #33579 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4⋊C8 ◁ C8×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
632 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6,10,14,11,16,26,15)(7,12,18,28,19,29,9,24)(8,20,21,27,23,35,22,36)(13,30,31,34,32,25,33,17);(2,3)(5,7,10,18,11,19,26,9)(6,12,14,28,16,29,15,24)(8,17,21,30,23,34,22,25)(13,27,31,35,32,36,33,20);(5,8)(6,13)(7,17)(9,25)(10,21)(11,23)(12,27)(14,31)(15,33)(16,32)(18,30)(19,34)(20,24)(22,26)(28,35)(29,36);(1,3,4,2)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=e,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=de,f^c=fh,h^c=hi110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405508);