Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483807 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
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 | C2×#3414 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1306 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)(5,6,10,17,14,8,9,27,12,19,29,16,11,18,28,15)(7,20,25,31,30,36,35,21,22,33,24,13,26,32,23,34);(1,2,3,4)(5,7,14,26,12,22,11,30)(6,13,18,34,19,31,8,21)(9,24,29,35,28,25,10,23)(15,32,16,20,27,36,17,33);(5,8)(6,14)(7,21)(9,27)(10,16)(11,19)(12,18)(13,26)(15,28)(17,29)(20,23)(22,34)(24,36)(25,32)(30,31)(33,35);(7,22)(13,31)(20,33)(21,34)(23,35)(24,25)(26,30)(32,36)Polycyclic presentation:
2^9|a^2=g,b^2=egh,e^2=h,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,d^b=di,e^b=eh,f^b=fhi,g^b=gh,h^b=hi,e^c=ehi,f^c=fh,g^c=ghi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72249996, #72250003, #72250341, … |
| maximal normal subgroup of | #72249996, #72250003, #72250341, … |
| maximal quotient of | #72285812, #72285928, #72287017, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401762);