Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #233501 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#127 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1314 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,10,14)(7,15,18,19)(8,11,9,13)(12,17,20,16)(21,22)(23,30)(24,28)(25,27)(26,32)(29,34)(31,36)(33,35);(2,4)(6,11)(8,9)(12,19)(13,14)(15,20)(16,17)(21,23,24,32)(22,26,28,30)(25,31,34,33)(27,35,29,36);(5,7)(6,12)(8,16)(9,17)(10,18)(11,19)(13,15)(14,20)(22,27)(23,31)(25,34)(28,29)(32,33)(35,36);(5,8,10,9)(6,11,14,13)(7,16,18,17)(12,19,20,15)(22,28)(26,30)(27,29)(35,36)Polycyclic presentation:
2^9|a^2=h,b^2=hi,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dhi,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,d^b=dh,e^b=eh,f^b=fi,g^b=ghi,e^c=eh,f^c=fi,g^c=ghi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 140708);