Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129430 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #410 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| socle | C22 | #1799 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
818 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)(7,16)(8,20)(9,14,23,12)(10,15)(11,17)(13,22)(18,30,34,29)(19,31)(21,27,35,28)(24,32,25,36)(26,33);(1,2)(3,4)(5,7,8,17,10,19,22,33)(6,11,13,16,15,26,20,31)(9,18,21,32,23,34,35,36)(12,24,27,29,14,25,28,30);(2,4)(6,12)(9,23)(11,24)(13,27)(14,15)(16,29)(18,34)(20,28)(21,35)(25,26)(30,31)(32,36)Polycyclic presentation:
2^9|a^2=g,b^2=d,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,h^a=hi,c^b=cf,e^c=ei,g^c=ghi,g^e=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55211);