Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #333055 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2669 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#459 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
674 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,8,12,7,3,6,11)(4,9,15,16,10,5,14,13)(17,18,22,26)(19,27,23,24)(20,30,21,31)(25,29,32,28);(1,3,7,2)(4,13,10,16)(5,14,9,15)(6,11,8,12)(17,19,22,23)(18,24,26,27)(20,28,21,29)(25,30,32,31);(1,4)(2,9)(3,5)(6,14)(7,10)(8,15)(11,13)(12,16)(17,20,22,21)(18,25,26,32)(19,28,23,29)(24,30,27,31);(1,5)(2,10)(3,4)(6,13)(7,9)(8,16)(11,14)(12,15)(17,21,22,20)(18,25,26,32)(19,29,23,28)(24,30,27,31)Polycyclic presentation:
2^9|a^2=eh,b^2=hi,c^2=i,d^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=defh,f^a=fi,c^b=cg,d^b=dei,e^b=eh,g^b=gh,f^c=fi,g^c=gh,e^d=eh,f^d=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 240318);