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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1914 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)(3,6)(4,8)(5,7)(9,10,13,14)(11,12,15,16)(17,18)(19,21)(20,22)(23,24)(25,26,31,32)(27,28,30,29);(1,3)(2,6)(4,7)(5,8)(9,10,13,14)(11,12,15,16)(18,19)(20,24)(25,27)(26,29)(28,32)(30,31);(2,7)(3,8)(10,14)(11,15)(17,18,21,19)(20,23,24,22)(25,28,31,29)(26,27,32,30);(1,3)(2,4)(5,8)(6,7)(9,11)(10,12)(13,15)(14,16)(17,19,21,18)(20,23,24,22)(25,29,31,28)(26,27,32,30);(1,4)(2,3)(5,6)(7,8)(9,12)(10,11)(13,16)(14,15)(17,20,21,24)(18,22,19,23)(25,30,31,27)(26,28,32,29)Polycyclic presentation:
2^9|a^2=gi,b^2=hi,c^2=f,d^2=f,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, 5143905);