Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4122169 |
| 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 | |
2240 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,6)(7,8)(9,10,13,12)(11,14,16,15)(17,18,21,23)(19,24,22,20)(25,26,30,29)(27,28,32,31);(1,2)(3,8)(4,7)(5,6)(10,12)(14,15)(17,19,21,22)(18,20,23,24)(26,29)(28,31);(1,3,5,4)(2,7,6,8)(9,11,13,16)(10,14,12,15)(17,18,21,23)(19,20,22,24)(25,27,30,32)(26,31,29,28);(1,4,5,3)(2,8,6,7)(9,12,13,10)(11,15,16,14)(17,20,21,24)(18,22,23,19)(25,28)(26,27)(29,32)(30,31);(1,2)(3,8)(4,7)(5,6)(9,11)(10,15)(12,14)(13,16)(17,18)(19,20)(21,23)(22,24)(25,29,30,26)(27,31,32,28)Polycyclic presentation:
2^9|a^2=fh,b^2=gi,c^2=f,d^2=fg,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4033278);