Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #234006 |
| 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×#3214 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1434 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,11)(8,16)(9,13)(10,15)(12,18)(14,20)(17,19)(21,22)(23,31)(24,29)(25,28)(26,30)(27,32)(33,36)(34,35);(1,2)(3,4)(5,7,9,14)(6,10,13,19)(8,17,18,15)(11,12,20,16)(21,23,26,34)(22,27,30,36)(24,33,25,32)(28,35,29,31);(2,4)(5,7)(6,11)(8,15)(9,14)(10,16)(12,19)(13,20)(17,18)(22,28)(24,25)(27,35)(29,30)(31,36)(32,33);(7,14)(10,19)(11,20)(15,17)(21,24,26,25)(22,29,30,28)(23,32,34,33)(27,35,36,31)Polycyclic presentation:
2^9|b^2=h,d^2=i,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=cg,d^b=dh,e^b=eh,f^b=fi,d^c=dh,e^c=ei,f^c=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 143065);