Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #504384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #3109 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #3054 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #1753 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
535 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,13)(7,17,19,35)(8,20,23,34)(9,24,25,36)(11,27,21,16)(12,28,29,18)(14,32,30,22)(15,33,31,26);(1,2)(3,4)(5,7)(6,12)(8,21)(9,22)(10,19)(11,23)(13,29)(14,16)(15,34)(17,24)(18,26)(20,31)(25,32)(27,30)(28,33)(35,36);(2,4)(6,13)(9,25)(12,29)(14,30)(16,27)(18,28)(22,32)(26,33);(5,8,11,7,10,23,21,19)(6,14,16,29,13,30,27,12)(9,18,26,32,25,28,33,22)(15,35,24,34,31,17,36,20)Polycyclic presentation:
2^9|a^2=i,d^2=fhi,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,d^b=dh,e^b=eh,f^b=fi,h^b=hi,e^c=ei,e^d=ehi,f^d=fi,h^d=hi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72321457 |
| maximal normal subgroup of | #72321457 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 411583);