Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| 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 | |
1854 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,8)(3,5,4,7)(11,15)(13,16)(17,18,21,22)(19,20,23,24)(27,31)(29,32);(1,2)(3,5)(4,7)(6,8)(9,10,12,14)(11,13,15,16)(17,19,21,23)(18,20,22,24);(1,3,6,4)(2,7,8,5)(9,11)(10,13)(12,15)(14,16)(17,20)(18,19)(21,24)(22,23)(25,26,28,30)(27,29,31,32);(1,4,6,3)(2,7,8,5)(9,10)(11,13)(12,14)(15,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,29)(28,31)(30,32);(1,5,6,7)(2,3,8,4)(9,11)(10,13)(12,15)(14,16)(17,18)(19,20)(21,22)(23,24)Polycyclic presentation:
2^9|a^2=h,b^2=fg,c^2=fi,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2106727);