Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #233469 |
| 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 | |
1354 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)(5,6,9,14)(7,15,17,19)(8,12,10,13)(11,18,20,16)(21,22)(23,26)(24,27)(25,28)(29,33)(30,35)(31,36)(32,34);(2,4)(5,7,9,17)(6,11,14,20)(8,18,10,16)(12,15,13,19)(21,23)(22,26)(24,30)(25,32)(27,35)(28,34)(29,36)(31,33);(5,6)(7,15)(8,12)(9,14)(10,13)(11,16)(17,19)(18,20)(22,27)(23,30)(24,33)(28,29)(31,32)(35,36);(5,8,9,10)(6,12,14,13)(7,16,17,18)(11,19,20,15)(22,28)(26,34)(27,29)(35,36)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=di,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, 140672);