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×#127 | C23 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
657 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,10)(3,11,7,8)(4,14,5,15)(9,13,16,12)(17,18,21,25,23,27,22,26)(19,28,29,20,31,32,30,24);(1,3,6,7)(2,8,10,11)(4,12,5,13)(9,14,16,15)(17,18)(19,28)(20,30)(21,26)(22,25)(23,27)(24,29)(31,32);(1,4,6,5)(2,9,10,16)(3,12,7,13)(8,14,11,15)(17,19)(18,24)(20,27)(21,29)(22,30)(23,31)(25,28)(26,32);(1,5,6,4)(2,9,10,16)(3,13,7,12)(8,14,11,15)(17,20,23,24)(18,19,27,31)(21,28,22,32)(25,30,26,29)Polycyclic presentation:
2^9|a^2=e,b^2=hi,c^2=hi,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, 240321);