Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33830 |
| 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 | #33716 | 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 | |
660 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,12,19,13,15,10,17,11,18,27,34,8,9,16,28)(7,20,25,14,21,33,29,26,24,22,35,32,30,36,23,31);(5,7,8,21,11,24,13,30)(6,14,9,26,18,32,15,31)(10,25,12,29,16,35,27,23)(17,33,19,22,28,36,34,20);(1,2,3,4)(5,8,11,13)(6,15,18,9)(7,21,24,30)(10,12,16,27)(14,31,32,26)(17,34,28,19)(20,36,22,33)(23,25,29,35);(5,9,27,17,13,6,16,34,11,15,12,28,8,18,10,19)(7,22,25,32,21,36,29,31,24,20,35,14,30,33,23,26)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=f,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=hi,d^c=di86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405750);