Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13404 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2034 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
486 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,3,4)(6,13)(7,18)(8,11)(9,24)(10,23)(14,31)(15,16)(17,29)(19,30)(20,26)(21,28)(22,35)(32,36)(33,34);(5,6,8,13,12,17,11,29)(7,19,20,34,22,36,21,31)(9,25,23,15,10,27,24,16)(14,26,32,18,33,28,30,35);(5,7)(6,14,17,33)(8,21)(9,26)(10,28)(11,20)(12,22)(13,30,29,32)(15,34,16,31)(18,24)(19,25,36,27)(23,35)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=dh,d^2=i,e^2=hi,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,f^a=fh,h^a=hi,c^b=cf,e^b=ehi,f^b=fhi,h^b=hi,d^c=di,e^c=eh,f^c=fhi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72321483, #72321498, #72321505, … |
| maximal normal subgroup of | #72321483, #72321498, #72321505, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56005);