Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4122169 |
| automorphism | Aut(G) order 1048576 |
| 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 | |
1977 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,4)(3,7)(9,10,13,12)(11,15,16,14)(17,18,21,24)(19,22,20,23)(26,29)(27,31);(1,2,5,4)(3,8,7,6)(9,10)(11,15)(12,13)(14,16)(17,19,21,20)(18,22,24,23)(26,29)(30,32);(2,4)(6,8)(10,12)(14,15)(17,19,21,20)(18,23,24,22)(25,26)(27,30)(28,29)(31,32);(1,3)(2,6)(4,8)(5,7)(9,11,13,16)(10,14,12,15)(17,20,21,19)(18,22,24,23)(25,27,28,31)(26,30,29,32);(1,4,5,2)(3,6,7,8)(9,12,13,10)(11,14,16,15)(17,18)(19,23)(20,22)(21,24)(25,26)(27,30)(28,29)(31,32)Polycyclic presentation:
2^9|a^2=fgh,b^2=gh,c^2=hi,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4038150);