Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | C22×#120 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2612 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,5)(6,8)(9,10,13,14)(11,15,16,12)(18,20)(19,23)(25,26,29,30)(27,28,31,32);(1,3)(2,6)(4,7)(5,8)(9,11,13,16)(10,12,14,15)(17,18,21,20)(19,22,23,24)(26,30)(28,32);(2,4)(6,7)(11,16)(12,15)(17,19)(18,22)(20,24)(21,23)(25,27,29,31)(26,28,30,32);(1,3)(2,7)(4,6)(5,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,24)(21,23)(25,28,29,32)(26,27,30,31);(1,4,5,2)(3,7,8,6)(9,12,13,15)(10,11,14,16)(17,20,21,18)(19,22,23,24)(25,27)(26,28)(29,31)(30,32)Polycyclic presentation:
2^9|a^2=fg,b^2=fghi,c^2=fgh,d^2=fgh,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3304408);