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 | |
1951 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)(3,7)(4,5)(6,8)(9,10)(11,12)(13,15)(14,16)(17,18,21,23)(19,24,22,20)(25,26,30,32)(27,28,31,29);(1,3,6,4)(2,5,8,7)(9,11,13,16)(10,14,15,12)(17,19)(18,20)(21,22)(23,24)(25,27,30,31)(26,28,32,29);(1,3,6,4)(2,7,8,5)(9,10)(11,12)(13,15)(14,16)(17,20)(18,22)(19,23)(21,24)(25,28,30,29)(26,27,32,31);(1,4,6,3)(2,7,8,5)(9,10)(11,14)(12,16)(13,15)(17,20)(18,19)(21,24)(22,23)(25,26)(27,28)(29,31)(30,32);(1,5,6,7)(2,4,8,3)(9,12,13,14)(10,11,15,16)(17,18)(19,24)(20,22)(21,23)(25,29,30,28)(26,31,32,27)Polycyclic presentation:
2^9|a^2=g,b^2=fg,c^2=i,d^2=fh,e^2=fg,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, 1372804);