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) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6,10,14,11,17,8,16)(7,18,22,29,19,31,20,12)(9,24,26,15,27,13,23,32)(21,30,36,34,25,33,35,28);(1,2,4,3)(5,7)(6,12,17,29)(8,20)(9,25)(10,22)(11,19)(13,28,24,34)(14,18,16,31)(15,33,32,30)(21,27)(23,36)(26,35);(1,3,4,2)(6,13)(8,10)(9,23)(12,28)(14,15)(16,32)(17,24)(18,33)(20,22)(21,35)(25,36)(26,27)(29,34)(30,31);(7,19)(12,29)(18,31)(20,22)(21,25)(28,34)(30,33)(35,36)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=eh,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 | #72319775, #72319799, #72319869, … |
| maximal normal subgroup of | #72319775, #72319799, #72319869, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402088);