Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #233265 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C24 | #121 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
746 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,14,11,16,10,15)(7,13,17,12,8,19,18,20)(21,22,25,30,26,29,35,36)(23,27,31,34,28,24,32,33);(1,2)(3,4)(5,7,9,17,11,8,10,18)(6,12,15,13,16,20,14,19)(21,23)(22,27)(24,29)(25,31)(26,28)(30,34)(32,35)(33,36);(2,4)(5,8,11,7)(6,13,16,19)(9,17,10,18)(12,14,20,15)(21,24,26,27)(22,28,29,23)(25,34,35,33)(30,31,36,32);(5,9,11,10)(6,14,16,15)(7,17,8,18)(12,19,20,13)(22,29)(23,28)(30,36)(31,32)Polycyclic presentation:
2^9|a^2=ef,b^2=e,c^2=hi,d^2=h,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,g^a=gh,c^b=cg,d^b=di,d^c=dh,e^c=eh,f^c=fi,g^c=gh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 149822);