Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4122169 |
| automorphism | Aut(G) order 3145728 |
| 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 | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2007 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,7,8,6)(9,10,13,12)(11,15,16,14)(17,18,21,22)(19,23,24,20)(26,27)(28,32);(2,5)(6,7)(10,12)(11,16)(17,19,21,24)(18,20,22,23)(28,32)(30,31);(3,8)(6,7)(10,12)(14,15)(18,22)(19,24)(25,26,29,27)(28,30,32,31);(1,2)(3,7)(4,5)(6,8)(9,11,13,16)(10,14,12,15)(17,20)(18,19)(21,23)(22,24)(25,27,29,26)(28,31,32,30);(1,3,4,8)(2,6,5,7)(9,12,13,10)(11,14,16,15)(17,19)(18,23)(20,22)(21,24)(25,28)(26,30)(27,31)(29,32)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=fhi,d^2=gh,e^2=fh,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, 4038411);