Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511896 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #3109 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
964 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)(3,4)(5,6,12,18)(7,14,20,34)(8,24,21,13)(9,25,19,16)(10,28,27,15)(11,30,26,17)(22,33,31,35)(23,32,29,36);(8,21)(9,19)(10,27)(14,34)(15,28)(16,25)(22,31)(32,36);(2,4)(5,7,11,29,12,20,26,23)(6,13,17,35,18,24,30,33)(8,19,31,10,21,9,22,27)(14,15,32,25,34,28,36,16);(5,8)(6,14)(7,19)(9,20)(10,29)(11,31)(12,21)(13,15)(16,33)(17,32)(18,34)(22,26)(23,27)(24,28)(25,35)(30,36)Polycyclic presentation:
2^9|a^2=i,c^2=eh,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=ghi,h^a=hi,d^b=di,f^b=fi,g^b=gi,e^c=ei,f^c=fhi,g^c=gh,h^c=hi,e^d=ei,f^d=fh,g^d=ghi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72250058, #72250074, #72250096 |
| maximal normal subgroup of | #72250058, #72250074, #72250096 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419311);