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) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15257 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2768 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,3)(4,8)(9,10)(11,15)(12,13)(14,16)(18,19)(20,24)(25,26,29,31)(27,32,30,28);(1,2,5,3)(4,7,8,6)(9,11,13,16)(10,14,12,15)(18,19)(22,23)(25,27,29,30)(26,28,31,32);(2,3)(6,7)(10,12)(14,15)(17,18,21,19)(20,22,24,23)(25,28,29,32)(26,30,31,27);(1,3,5,2)(4,7,8,6)(9,12,13,10)(11,14,16,15)(17,19,21,18)(20,23,24,22)(25,28)(26,27)(29,32)(30,31);(1,4)(2,6)(3,7)(5,8)(9,11)(10,15)(12,14)(13,16)(17,20)(18,22)(19,23)(21,24)(25,27)(26,28)(29,30)(31,32)Polycyclic presentation:
2^9|a^2=fgh,b^2=f,c^2=fi,d^2=i,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, 4423811);