Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13331 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #459 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1982 |
| prime radical | C2×#3147 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
1162 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6)(7,18,21,34)(8,23)(9,26)(10,28)(11,16)(12,19,31,35)(13,20,33,36)(14,24)(15,25)(17,27)(22,30,29,32);(2,4)(5,7)(6,12)(8,20)(9,19)(10,29)(11,21)(13,28)(14,30)(15,18)(16,31)(17,33)(22,27)(23,34)(24,35)(25,36)(26,32);(5,7)(6,13)(8,20)(9,19)(10,29)(11,21)(12,17)(14,18)(15,32)(16,33)(22,27)(23,30)(24,35)(25,36)(26,34)(28,31)Polycyclic presentation:
2^9|a^2=g,d^2=h,e^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fi,h^a=hi,c^b=cf,d^b=dhi,e^b=eh,g^b=gi,h^b=hi,d^c=dhi,e^c=eh,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55063);