Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33539 |
| 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 | |
1052 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(6,13)(7,18)(8,11)(9,23)(10,22)(14,30)(15,16)(17,26)(19,20)(21,25)(24,27)(28,35)(29,32)(33,34);(1,3,4,2)(5,6)(7,14)(8,15)(9,24)(10,26)(11,27)(12,17)(13,23)(16,22)(18,31)(19,30)(20,32)(21,34)(25,36)(28,33)(29,35);(5,7)(6,14)(8,19)(9,25)(10,20)(11,28)(12,21)(13,29)(15,30)(16,31)(17,34)(18,22)(23,35)(24,36)(26,32)(27,33);(5,8,12,11)(6,15,17,27)(7,19,21,28)(9,10,22,23)(13,24,26,16)(14,30,34,33)(18,35,25,20)(29,36,32,31)Polycyclic presentation:
2^9|a^2=g,b^2=g,d^2=i,e^2=hi,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fhi,h^a=hi,e^b=eh,f^b=fhi,h^b=hi,e^c=ehi,f^c=fhi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72321498, #72321536, #72321598, … |
| maximal normal subgroup of | #72321498, #72321536, #72321598, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401511);