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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1658 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,3)(4,6,8,7)(9,10)(11,14)(12,13)(15,16)(18,20)(19,23);(2,3)(6,7)(9,10,13,12)(11,14,16,15)(19,23)(22,24)(25,26,28,30)(27,29,31,32);(4,8)(6,7)(10,12)(14,15)(17,18,21,20)(19,22,23,24)(25,26)(27,29)(28,30)(31,32);(1,3,5,2)(4,7,8,6)(9,11,13,16)(10,14,12,15)(17,19)(18,22)(20,24)(21,23);(1,4,5,8)(2,6,3,7)(9,12,13,10)(11,15,16,14)(17,20,21,18)(19,24,23,22)(25,27,28,31)(26,29,30,32)Polycyclic presentation:
2^9|a^2=fgi,b^2=h,c^2=i,d^2=f,e^2=fgh,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, 2148194);