Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13504 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2077 |
| 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 | |
450 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,10,17,11,18,29,15,12,19,30,16,14,8,9,28)(7,20,24,33,25,21,36,35,26,34,23,13,31,32,22,27);(1,2)(3,4)(5,7,11,31,12,26,14,25)(6,13,18,35,19,33,8,27)(9,24,10,22,29,23,30,36)(15,21,16,20,28,32,17,34);(2,4)(5,8)(6,14)(7,21,26,32)(9,15)(10,17)(11,19)(12,18)(13,23,33,24)(16,30)(20,31,34,25)(22,27,36,35)(28,29)Polycyclic presentation:
2^9|a^2=g,b^2=deh,c^2=de,d^2=hi,e^2=h,f^2=h,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,c^b=cf,d^b=dh,e^b=eh,f^b=fh,g^b=ghi,h^b=hi,d^c=dh,e^c=ehi,f^c=fh,g^c=gh,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72239848, #72239855, #72240855, … |
| maximal normal subgroup of | #72239848, #72239855, #72240855, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56593);