Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2813 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,16)(12,15)(13,14)(18,22)(19,20)(25,26,30,32)(27,28,31,29);(1,2,4,6)(3,5,8,7)(10,14)(15,16)(17,18,21,22)(19,24,20,23)(25,27)(26,29)(28,32)(30,31);(1,2)(3,5)(4,6)(7,8)(9,11,13,12)(10,15,14,16)(17,19,21,20)(18,23,22,24)(25,28,30,29)(26,31,32,27);(1,2)(3,5)(4,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,23)(20,24)(21,22)(25,27)(26,28)(29,32)(30,31);(1,3)(2,5)(4,8)(6,7)(9,12,13,11)(10,15,14,16)(17,20,21,19)(18,23,22,24)(25,29,30,28)(26,31,32,27)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4776409);