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) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1901 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)(3,5)(4,7)(6,8)(9,10,13,14)(11,15,16,12)(17,18,21,24)(19,20,23,22)(25,26,30,32)(27,28,31,29);(2,3)(6,7)(9,11,13,16)(10,12,14,15)(17,19,21,23)(18,22,24,20)(25,27)(26,29)(28,32)(30,31);(1,2,5,3)(4,6,8,7)(9,10)(11,12)(13,14)(15,16)(17,20)(18,23)(19,24)(21,22)(25,28,30,29)(26,31,32,27);(1,3,5,2)(4,6,8,7)(9,10)(11,15)(12,16)(13,14)(17,20)(18,19)(21,22)(23,24)(25,27)(26,28)(29,32)(30,31);(1,4,5,8)(2,6,3,7)(9,12,13,15)(10,11,14,16)(17,18)(19,22)(20,23)(21,24)(25,29,30,28)(26,31,32,27)Polycyclic presentation:
2^9|a^2=gh,b^2=fghi,c^2=ghi,d^2=fgh,e^2=fgi,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, 4835942);