Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4033837 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 | |
2266 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)(10,14)(12,15)(17,18,23,26)(19,24,20,22)(21,29,31,25)(27,32,28,30);(6,8)(10,14)(12,15)(17,19,23,20)(18,22,26,24)(21,28,31,27)(25,30,29,32);(5,6,7,8)(9,10)(11,15)(12,16)(13,14)(17,20,23,19)(18,24,26,22)(21,28,31,27)(25,30,29,32);(1,2,3,4)(9,11,13,16)(10,15,14,12)(17,21)(18,25)(19,27)(20,28)(22,32)(23,31)(24,30)(26,29);(9,12,13,15)(10,16,14,11)(17,22,23,24)(18,20,26,19)(21,30,31,32)(25,27,29,28)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=gh,d^2=gi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4007678);