Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2601293 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2576 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,3,4)(10,14)(11,16)(17,18,22,19)(20,23,28,25)(21,24,30,26)(27,31,29,32);(6,8)(9,10,13,14)(11,12,16,15)(18,19)(23,25)(24,26)(31,32);(5,6)(7,8)(9,11)(10,15)(12,14)(13,16)(17,19,22,18)(20,25,28,23)(21,26,30,24)(27,32,29,31);(1,2)(3,4)(9,12,13,15)(10,11,14,16)(17,20,22,28)(18,23,19,25)(21,29,30,27)(24,32,26,31);(9,11)(10,12)(13,16)(14,15)(17,21,22,30)(18,24,19,26)(20,27,28,29)(23,31,25,32)Polycyclic presentation:
2^9|a^2=fgi,b^2=g,c^2=fg,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2514054);