Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483807 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
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 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
794 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6,8,14,11,17,9,16)(7,18,21,32,22,30,19,12)(10,25,23,15,28,35,24,29)(13,27,34,20,31,36,33,26);(1,2,4,3)(5,7)(6,12,17,32)(8,21)(9,19)(10,26)(11,22)(13,29,31,15)(14,18,16,30)(20,28)(23,27)(24,36)(25,33,35,34);(1,3,4,2)(5,7)(6,13)(8,19)(9,21)(10,27)(11,22)(12,29)(14,33)(15,32)(16,34)(17,31)(18,35)(20,24)(23,26)(25,30)(28,36);(5,8,11,9)(6,14,17,16)(7,19,22,21)(10,23,28,24)(12,30,32,18)(13,33,31,34)(15,35,29,25)(20,27,26,36)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=eh,d^2=i,e^2=i,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,d^b=di,f^b=fi,e^c=ei,f^c=fhi,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72239529, #72239538, #72239548, … |
| maximal normal subgroup of | #72239529, #72239538, #72239548, … |
| maximal quotient of | #72319150, #72319186, #72320064, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402089);