Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483873 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1048 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)(5,6,9,13)(7,15)(8,20,22,33)(10,25,24,14)(11,26)(12,30,32,23)(16,34)(17,29)(18,31)(19,35)(21,27)(28,36);(5,7)(6,11)(8,21)(9,19)(10,16)(12,31)(13,29)(14,27)(15,33)(17,30)(18,24)(20,35)(22,36)(23,26)(25,28)(32,34);(2,4);(7,16,19,18)(8,22)(11,27,29,28)(12,32)(15,34,35,31)(17,36,26,21)(20,33)(23,30)Polycyclic presentation:
2^9|a^2=g,d^2=gi,e^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=dh,e^b=ehi,g^b=gi,h^b=hi,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72249903, #72250058, #72250062, … |
| maximal normal subgroup of | #72249903, #72250058, #72250062, … |
| maximal quotient of | #72285406, #72285489, #72285836, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402918);