Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #233019 |
| 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 | |
716 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,13,10,14,8,17)(7,11,15,19,16,20,18,12)(21,22,26,28)(23,30,32,34)(24,31,27,36)(25,33,35,29);(1,2)(3,4)(5,7)(6,11)(8,18)(9,15)(10,16)(12,17)(13,19)(14,20)(21,23,25,31,26,32,35,36)(22,27,29,30,28,24,33,34);(2,4)(5,7)(6,12)(8,15)(9,18)(10,16)(11,17)(13,20)(14,19)(21,24,26,27)(22,23,28,32)(25,30,35,34)(29,36,33,31);(5,8,10,9)(6,13,14,17)(7,15,16,18)(11,12,20,19)(22,28)(24,27)(29,33)(30,34)Polycyclic presentation:
2^9|a^2=ef,b^2=e,c^2=h,d^2=hi,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=dhi,d^c=dh,e^c=eh,f^c=fi,g^c=gh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 149860);