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 | |
2099 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,15)(12,14)(13,16)(17,18)(19,23)(20,24)(21,22)(26,27)(28,32);(2,3)(4,8)(9,10,12,14)(11,13,16,15)(17,19,21,24)(18,20,22,23)(26,27)(30,31);(2,3)(6,7)(11,16)(13,15)(18,22)(20,23)(25,26,29,27)(28,30,32,31);(1,3,5,2)(4,7,8,6)(9,11)(10,13)(12,16)(14,15)(17,20)(18,19)(21,23)(22,24)(25,27,29,26)(28,31,32,30);(1,4)(2,6)(3,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,20)(21,24)(22,23)(25,28,29,32)(26,30,27,31)Polycyclic presentation:
2^9|a^2=fhi,b^2=hi,c^2=h,d^2=fi,e^2=h,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, 1923981);