Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13373 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #486 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #2023 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
482 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,10,17,11,18,8,16,12,19,27,30,9,15,28,14)(7,20,24,13,23,34,22,32,25,33,26,35,31,21,29,36);(1,2)(3,4)(5,7,9,23,12,25,11,31)(6,13,15,32,19,35,18,36)(8,26,10,29,28,24,27,22)(14,34,30,33,16,21,17,20);(2,4)(5,6,8,14,9,15,10,30,12,19,28,16,11,18,27,17)(7,21,22,35,31,33,24,32,25,34,29,13,23,20,26,36)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=d,d^2=hi,e^2=i,f^2=hi,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,d^b=dh,f^b=fh,g^b=ghi,h^b=hi,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250257, #72250266, #72250310, … |
| maximal normal subgroup of | #72250257, #72250266, #72250310, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56249);