Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#392 |
| automorphism | Aut(G) order 6291456 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1818 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,5)(4,8)(6,7)(9,10,13,14)(11,12,15,16)(17,18,22,23)(19,24,20,21)(26,31)(27,28);(1,3,6,4)(2,5,7,8)(9,10,13,14)(11,12,15,16)(18,23)(21,24)(25,26,30,31)(27,29,28,32);(2,7)(5,8)(10,14)(12,16)(17,19,22,20)(18,21,23,24)(25,27,30,28)(26,32,31,29);(1,4,6,3)(2,8,7,5)(9,11,13,15)(10,12,14,16)(17,20,22,19)(18,21,23,24)(25,28,30,27)(26,32,31,29);(1,5,6,8)(2,3,7,4)(9,12)(10,15)(11,14)(13,16)(17,21)(18,19)(20,23)(22,24)(25,29,30,32)(26,27,31,28)Polycyclic presentation:
2^9|a^2=fh,b^2=fgi,c^2=f,d^2=g,e^2=fgh,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^c=eg110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4214638);