Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #246847 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#121 | D4 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1073 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(6,12)(7,14)(8,17)(9,10)(16,18)(19,20)(21,22,27,24)(23,31,25,34)(26,33,36,30)(28,35,29,32);(1,2)(3,4)(5,6)(7,15)(8,11)(9,17)(10,12)(13,16)(14,19)(18,20)(21,23)(22,28)(24,29)(25,27)(26,35)(30,34)(31,33)(32,36);(2,4)(5,7,11,16)(6,13,8,15)(9,14,10,18)(12,19,17,20)(21,24,27,22)(23,29,25,28)(26,30,36,33)(31,35,34,32);(5,8)(6,11)(7,15)(9,12)(10,17)(13,16)(14,19)(18,20)(21,25,27,23)(22,29,24,28)(26,32,36,35)(30,31,33,34)Polycyclic presentation:
2^9|a^2=i,c^2=h,d^2=i,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fhi,c^b=cg,d^b=di,e^b=eh,f^b=fi,e^c=ei,f^c=fhi,e^d=eh,f^d=fi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 154098);