Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129430 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
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×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
831 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)(7,15)(8,19)(9,22,23,12)(10,14)(11,16)(13,21)(17,31,32,27)(18,28)(20,26,34,33)(24,29,35,36)(25,30);(1,2)(3,4)(5,7,8,16,10,18,21,30)(6,11,13,15,14,25,19,28)(9,17,20,29,23,32,34,36)(12,24,26,27,22,35,33,31);(2,4)(6,12)(11,24)(13,26)(14,22)(15,27)(19,33)(25,35)(28,31)Polycyclic presentation:
2^9|a^2=g,b^2=d,d^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,h^a=hi,c^b=cf,g^c=gh,g^e=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55209);