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:
(5,6,10,18,9,14,8,13)(7,11,15,12,17,20,16,19)(21,22,27,30)(23,26,29,25)(24,32,35,31)(28,33,36,34);(1,2)(3,4)(5,7)(6,11)(8,16)(9,17)(10,15)(12,18)(13,19)(14,20)(21,23,25,22,27,29,26,30)(24,28,33,31,35,36,34,32);(2,4)(5,7)(6,12)(8,15)(9,17)(10,16)(11,18)(13,20)(14,19)(21,24,27,35)(22,28,30,36)(23,31,29,32)(25,34,26,33);(5,8,9,10)(6,13,14,18)(7,15,17,16)(11,12,20,19)(21,25,27,26)(22,23,30,29)(24,33,35,34)(28,32,36,31)Polycyclic presentation:
2^9|a^2=efi,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, 149704);