Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1987 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)(10,14)(11,16)(17,18,21,22)(19,20,23,24)(27,31)(29,32);(1,2,5,4)(3,6,8,7)(9,10)(11,12)(13,14)(15,16)(17,19,21,23)(18,20,22,24);(2,4)(6,7)(10,14)(12,15)(17,20)(18,19)(21,24)(22,23)(25,26,28,30)(27,29,31,32);(1,3,5,8)(2,6,4,7)(9,11,13,16)(10,15,14,12)(17,19)(18,20)(21,23)(22,24)(25,27)(26,29)(28,31)(30,32);(1,4,5,2)(3,6,8,7)(9,12,13,15)(10,11,14,16)(17,18)(19,20)(21,22)(23,24)Polycyclic presentation:
2^9|a^2=fh,b^2=g,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=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3306566);