Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1258513 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
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 | |
1867 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,8)(3,4,7,6)(9,10,12,14)(11,15,16,13)(17,18,21,23)(19,24,22,20)(26,29)(27,31);(1,2)(3,6)(4,7)(5,8)(17,19,21,22)(18,20,23,24);(1,3)(2,4)(5,7)(6,8)(9,10)(11,13)(12,14)(15,16)(17,19,21,22)(18,20,23,24)(25,26,28,29)(27,30,31,32);(1,3)(2,6)(4,8)(5,7)(9,11,12,16)(10,13,14,15)(17,18)(19,20)(21,23)(22,24);(1,4,5,6)(2,7,8,3)(9,10)(11,13)(12,14)(15,16)(17,20)(18,22)(19,23)(21,24)(25,27)(26,30)(28,31)(29,32)Polycyclic presentation:
2^9|a^2=i,b^2=fh,c^2=fgi,d^2=fi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1233630);