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)(5,6,11,8)(7,15,17,16)(9,14,10,13)(12,19,20,18)(21,22,27,30)(23,33,29,36)(24,34,28,35)(25,32,26,31);(2,4)(5,7)(6,12)(8,20)(9,19)(10,18)(11,17)(13,16)(14,15)(21,23)(22,28)(24,30)(25,35)(26,34)(27,29)(31,36)(32,33);(5,8,11,6)(7,16,17,15)(9,13,10,14)(12,18,20,19)(21,24,27,28)(22,29,30,23)(25,33,26,36)(31,34,32,35);(7,17)(12,20)(15,16)(18,19)(22,30)(23,29)(31,32)(34,35)Polycyclic presentation:
2^9|a^2=h,c^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, 140668);