Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33531 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| 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 | |
1068 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6)(7,18)(8,20)(9,14)(10,27)(11,30)(12,33)(13,17)(15,26)(16,25)(19,32)(21,29)(22,35)(23,31)(24,36)(28,34);(1,3,4,2)(5,7)(6,8)(9,21)(10,28)(11,31)(12,19)(13,22)(14,34)(15,32)(16,29)(17,24)(18,27)(20,26)(23,33)(25,36)(30,35);(5,8)(6,7)(9,23)(10,29)(11,32)(12,34)(13,24)(14,19)(15,31)(16,28)(17,22)(18,25)(20,30)(21,33)(26,35)(27,36);(5,9,13,12)(6,14,17,33)(7,19,22,21)(8,23,24,34)(10,25,11,26)(15,27,16,30)(18,32,35,29)(20,31,36,28)Polycyclic presentation:
2^9|a^2=g,b^2=g,d^2=i,e^2=hi,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,h^a=hi,d^b=di,e^b=eh,f^b=fhi,h^b=hi,e^c=ehi,f^c=fh,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72250274, #72250290, #72250316, … |
| maximal normal subgroup of | #72250274, #72250290, #72250316, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401504);