Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| 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 | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2019 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)(3,5,7,8)(9,10)(11,16)(12,14)(13,15)(17,18,22,24)(19,21,20,23)(25,26)(27,31)(28,30)(29,32);(9,11,13,12)(10,14,15,16)(17,19)(18,23)(20,22)(21,24)(25,26,28,30)(27,29,32,31);(11,12)(14,16)(18,24)(19,20)(27,32)(29,31);(1,2)(3,5)(4,6)(7,8)(9,12,13,11)(10,14,15,16)(17,20,22,19)(18,21,24,23)(25,27)(26,29)(28,32)(30,31);(1,3,4,7)(2,5,6,8)(9,10)(11,16)(12,14)(13,15)(17,21)(18,19)(20,24)(22,23)(25,26)(27,29)(28,30)(31,32)Polycyclic presentation:
2^9|a^2=fhi,b^2=fg,d^2=h,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1886024);