Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33539 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
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×#509 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1354 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,10,15,27,14,29,16,12,17,26,28,9,7,11,19)(8,20,22,32,21,34,35,30,25,33,36,13,24,31,23,18);(1,2,3,4)(5,7)(6,9)(8,18)(10,28)(11,19)(12,14)(13,21)(15,26)(16,29)(17,27)(20,23)(22,31)(24,32)(25,30)(33,35)(34,36);(5,8)(6,13)(7,18)(9,21)(10,23)(11,22)(12,25)(14,30)(15,33)(16,34)(17,32)(19,31)(20,28)(24,27)(26,35)(29,36);(5,9,12,27)(6,14,17,7)(8,21,25,24)(10,11,26,29)(13,30,32,18)(15,16,28,19)(20,31,33,34)(22,35,36,23)Polycyclic presentation:
2^9|a^2=g,b^2=egh,d^2=i,e^2=hi,f^2=h,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,f^a=fi,e^b=eh,f^b=fh,g^b=gh,h^b=hi,e^c=eh,f^c=fhi,g^c=gh,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72250371, #72250374, #72250380, … |
| maximal normal subgroup of | #72250371, #72250374, #72250380, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401716);