Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1418353 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1918 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,5,8,7)(9,10)(11,13)(12,14)(15,16)(18,23)(19,21)(27,32)(28,30);(1,2)(3,7)(4,6)(5,8)(9,11)(10,12)(13,16)(14,15)(17,18,22,23)(19,24,21,20)(25,26,29,31)(27,30,32,28);(9,12,15,13)(10,11,16,14)(17,19,22,21)(18,24,23,20)(25,27)(26,30)(28,31)(29,32);(1,2)(3,7)(4,6)(5,8)(9,13,15,12)(10,11,16,14)(17,20)(18,19)(21,23)(22,24);(1,3)(2,5)(4,8)(6,7)(9,14,15,11)(10,13,16,12)(17,21,22,19)(18,24,23,20)(25,28,29,30)(26,27,31,32)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=fi,d^2=h,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,e^d=ef92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1392036);