Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12019 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#106 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C22×#106 |
| prime radical | C25 | SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1018 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,16,19,30)(8,13)(9,22)(10,25)(11,15)(12,18,27,32)(14,23)(17,29,20,34)(21,28)(24,35)(26,31,36,33);(5,7,9,18,10,20,23,33,11,19,24,32,8,17,21,31)(6,12,14,16,13,26,22,29,15,27,28,30,25,36,35,34);(2,4);(5,8,11,10)(6,13,15,25)(7,17,19,20)(9,21,24,23)(12,26,27,36)(14,22,28,35)(16,29,30,34)(18,31,32,33)Polycyclic presentation:
2^9|a^2=g,b^2=e,d^2=i,e^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,g^b=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72239487, #72239817, #72239831, … |
| maximal normal subgroup of | #72239487, #72239817, #72239831, … |
| maximal quotient of | #72285685, #72309618, #72310106, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400998);