Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1418353 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1984 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)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,22)(20,23)(21,24)(25,26,30,28)(27,32,29,31);(2,6)(4,7)(10,14)(12,15)(17,19,20,24)(18,21,23,22);(9,11)(10,15)(12,14)(13,16)(17,18)(19,22)(20,23)(21,24)(25,27,30,29)(26,31,28,32);(1,3,5,8)(2,7,6,4)(9,11)(10,12)(13,16)(14,15)(17,18)(19,21)(20,23)(22,24)(25,28,30,26)(27,32,29,31);(1,4,5,7)(2,3,6,8)(9,12)(10,16)(11,14)(13,15)(17,19)(18,22)(20,24)(21,23)(25,29,30,27)(26,32,28,31)Polycyclic presentation:
2^9|a^2=fghi,b^2=h,c^2=fghi,d^2=fgh,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,e^d=ef92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1401295);