Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #233100 |
| 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 | |
718 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,14,10,15,8,13)(7,11,16,19,17,20,18,12)(21,22,27,29)(23,26,31,25)(24,32,35,30)(28,33,36,34);(1,2)(3,4)(5,7)(6,11)(8,18)(9,16)(10,17)(12,13)(14,19)(15,20)(21,23,26,29,27,31,25,22)(24,28,34,32,35,36,33,30);(2,4)(5,7)(6,12)(8,16)(9,18)(10,17)(11,13)(14,20)(15,19)(21,24,27,35)(22,28,29,36)(23,30,31,32)(25,34,26,33);(5,8,10,9)(6,13,15,14)(7,16,17,18)(11,19,20,12)(21,25,27,26)(22,23,29,31)(24,33,35,34)(28,32,36,30)Polycyclic presentation:
2^9|a^2=ef,b^2=e,c^2=h,d^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,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, 149700);