Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1082705 |
| 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 | |
2168 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)(4,8)(5,6)(9,10)(11,15)(12,13)(14,16)(17,18,20,22)(19,21,24,23);(1,3,5,8)(2,4,6,7)(10,12)(14,15)(17,18)(19,23)(20,22)(21,24)(26,29)(30,32);(2,6)(3,8)(9,11,13,16)(10,14,12,15)(17,18,20,22)(19,21,24,23)(25,26,28,29)(27,30,31,32);(1,2)(3,4)(5,6)(7,8)(9,11)(10,14)(12,15)(13,16)(17,18)(19,21)(20,22)(23,24);(1,4)(2,3)(5,7)(6,8)(9,12,13,10)(11,15,16,14)(17,19,20,24)(18,21,22,23)(25,27,28,31)(26,30,29,32)Polycyclic presentation:
2^9|a^2=fg,b^2=fi,c^2=gh,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1032872);