Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15257 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2580 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,16)(13,15)(17,18,21,23)(19,20,24,22)(25,26,29,28)(27,31,32,30);(2,4)(3,7)(9,11)(10,14)(12,15)(13,16)(17,19,21,24)(18,22,23,20)(25,26)(27,31)(28,29)(30,32);(2,4)(6,8)(10,15)(11,16)(18,23)(20,22)(26,28)(30,31);(1,3)(2,6)(4,8)(5,7)(9,11)(10,12)(13,16)(14,15)(17,19)(18,22)(20,23)(21,24)(25,27)(26,30)(28,31)(29,32);(1,4,5,2)(3,8,7,6)(9,12)(10,11)(13,14)(15,16)(17,20,21,22)(18,19,23,24)(25,28,29,26)(27,31,32,30)Polycyclic presentation:
2^9|a^2=fgi,b^2=hi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5143900);