Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| automorphism | Aut(G) order 1048576 |
| 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 | #15124 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2655 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,3)(4,6,8,7)(9,10)(11,16)(12,15)(13,14)(17,18,21,20)(19,23,24,22)(27,31)(30,32);(2,3)(6,7)(9,11,14,12)(10,15,13,16)(17,19)(18,22)(20,23)(21,24);(4,8)(6,7)(10,13)(15,16)(19,24)(22,23)(25,26,29,28)(27,30,31,32);(1,3,5,2)(4,7,8,6)(9,12,14,11)(10,16,13,15)(17,20,21,18)(19,22,24,23)(25,27,29,31)(26,30,28,32);(1,4,5,8)(2,6,3,7)(9,13,14,10)(11,16,12,15)(17,19)(18,23)(20,22)(21,24)(25,28,29,26)(27,32,31,30)Polycyclic presentation:
2^9|a^2=fg,b^2=g,c^2=gi,d^2=fgi,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2501250);