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 | C26 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2428 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,6)(3,4,7,8)(9,10,12,14)(11,15,16,13)(17,18,22,23)(19,21,20,24)(27,31)(29,32);(2,6)(4,8)(9,11,12,16)(10,13,14,15)(18,23)(21,24)(25,26,28,30)(27,29,31,32);(2,6)(4,8)(11,16)(13,15)(17,19,22,20)(18,24,23,21)(25,27)(26,29)(28,31)(30,32);(1,3)(2,4)(5,7)(6,8)(9,10)(11,15)(12,14)(13,16)(17,20,22,19)(18,21,23,24)(25,26)(27,29)(28,30)(31,32);(1,4)(2,7)(3,6)(5,8)(9,11)(10,13)(12,16)(14,15)(17,21)(18,20)(19,23)(22,24)(25,27)(26,29)(28,31)(30,32)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=hi,d^2=hi,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, 3908748);