Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2805613 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2318 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,4)(9,10)(11,15)(12,16)(13,14)(17,18,22,26)(19,27,29,23)(20,24,30,32)(21,31,28,25);(6,8)(10,14)(12,15)(17,19,22,29)(18,23,26,27)(20,21,30,28)(24,25,32,31);(5,6,7,8)(17,18)(19,23)(20,24)(21,25)(22,26)(27,29)(28,31)(30,32);(1,2)(3,4)(9,11)(10,15)(12,14)(13,16)(17,20,22,30)(18,24,26,32)(19,28,29,21)(23,31,27,25);(9,12,13,15)(10,16,14,11)(17,21,22,28)(18,25,26,31)(19,20,29,30)(23,24,27,32)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=h,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2714931);