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 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2049 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,3)(4,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,23)(20,22)(21,24)(26,28)(31,32);(1,2,5,3)(4,7,8,6)(9,11,14,16)(10,13,12,15)(17,19,22,21)(18,23,20,24)(25,26,30,28)(27,31,29,32);(2,3)(6,7)(10,12)(11,16)(17,18,22,20)(19,24,21,23)(25,27,30,29)(26,31,28,32);(1,3,5,2)(4,7,8,6)(9,12,14,10)(11,13,16,15)(17,20,22,18)(19,23,21,24)(25,28,30,26)(27,31,29,32);(1,4)(2,6)(3,7)(5,8)(9,13)(10,11)(12,16)(14,15)(17,21,22,19)(18,23,20,24)(25,29,30,27)(26,31,28,32)Polycyclic presentation:
2^9|b^2=i,c^2=fh,d^2=i,e^2=fh,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, 4419656);