Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #845297 |
| 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 | |
2075 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(3,7)(6,8)(10,13)(14,16)(17,18,22,20)(19,24,21,23);(1,2,5,4)(3,6,7,8)(10,13)(14,16)(17,19,22,21)(18,23,20,24)(26,29)(30,32);(9,10)(11,14)(12,13)(15,16)(17,18,22,20)(19,23,21,24)(25,26)(27,30)(28,29)(31,32);(1,3)(2,6)(4,8)(5,7)(9,10)(11,14)(12,13)(15,16)(17,20,22,18)(19,24,21,23);(1,4,5,2)(3,8,7,6)(9,11,12,15)(10,14,13,16)(17,21,22,19)(18,24,20,23)(25,27,28,31)(26,30,29,32)Polycyclic presentation:
2^9|a^2=f,b^2=fgi,c^2=f,d^2=f,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 805920);