Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4033837 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2078 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,7,8,6)(10,13)(14,16)(17,18)(19,22)(20,24)(21,23)(25,26)(27,30)(28,29)(31,32);(2,5)(3,8)(9,10,12,13)(11,14,15,16)(17,19,21,24)(18,22,23,20)(25,27,29,32)(26,30,28,31);(2,5)(6,7)(11,15)(14,16)(17,19)(18,20)(21,24)(22,23)(25,28,29,26)(27,30,32,31);(1,3)(2,6)(4,8)(5,7)(9,10)(11,16)(12,13)(14,15)(17,20)(18,19)(21,22)(23,24);(1,2)(3,7)(4,5)(6,8)(9,11)(10,14)(12,15)(13,16)(17,19)(18,20)(21,24)(22,23)(25,27)(26,31)(28,30)(29,32)Polycyclic presentation:
2^9|a^2=gh,b^2=fg,c^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3971167);