Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1258513 |
| automorphism | ? |
| 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 | |
2103 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,5)(3,8,4,7)(9,10,13,16)(11,14,15,12)(17,18,21,23)(19,24,20,22)(25,26,30,27)(28,32,29,31);(1,3,6,4)(2,7,5,8)(9,11,13,15)(10,14,16,12)(17,19,21,20)(18,22,23,24)(25,27,30,26)(28,32,29,31);(9,10)(11,14)(12,15)(13,16)(17,18)(19,24)(20,22)(21,23)(25,28,30,29)(26,31,27,32);(1,4,6,3)(2,7,5,8)(9,10)(11,14)(12,15)(13,16)(17,20,21,19)(18,22,23,24);(1,5,6,2)(3,8,4,7)(9,12,13,14)(10,15,16,11)(17,18)(19,22)(20,24)(21,23)(25,29,30,28)(26,32,27,31)Polycyclic presentation:
2^9|a^2=fgh,b^2=fgh,c^2=ghi,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1194380);