Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3858029 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1792 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)(3,6)(4,5)(7,8)(9,10)(11,16)(12,14)(13,15)(17,18,21,22)(19,20,23,24)(25,26)(27,31)(28,32)(29,30);(2,5)(6,7)(9,11,13,12)(10,14,15,16)(17,18)(19,20)(21,22)(23,24)(26,29)(27,28);(2,5)(6,7)(10,15)(14,16)(18,22)(20,24)(25,27,30,28)(26,31,29,32);(1,3)(2,6)(4,8)(5,7)(9,12,13,11)(10,16,15,14)(17,19)(18,20)(21,23)(22,24)(25,28,30,27)(26,32,29,31);(1,2)(3,7)(4,5)(6,8)(9,10)(11,16)(12,14)(13,15)(17,20)(18,23)(19,22)(21,24)(25,29,30,26)(27,31,28,32)Polycyclic presentation:
2^9|a^2=fgh,b^2=i,c^2=fg,d^2=fgi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3870438);