Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2539 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,7)(3,8,4,6)(9,10)(11,15)(12,13)(14,16)(17,18)(19,23)(20,22)(21,24)(25,26)(27,29)(28,30)(31,32);(1,3,5,4)(2,6,7,8)(9,11,13,16)(10,14,12,15)(17,18,20,22)(19,21,24,23)(25,26,28,30)(27,31,32,29);(3,4)(6,8)(10,12)(14,15)(19,24)(21,23)(25,26,28,30)(27,29,32,31);(1,4,5,3)(2,6,7,8)(9,12,13,10)(11,14,16,15)(17,19)(18,21)(20,24)(22,23)(25,26)(27,29)(28,30)(31,32);(1,2)(3,8)(4,6)(5,7)(9,11)(10,15)(12,14)(13,16)(17,18)(19,21)(20,22)(23,24)(25,27)(26,29)(28,32)(30,31)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=fhi,d^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4428138);