Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1354 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(11,15)(13,16)(17,18,21,19)(20,23,24,22)(25,26,30,31)(27,32,33,28)(29,34,38,39)(35,37,40,36);(3,7)(5,8)(9,10,12,14)(11,13,15,16)(17,18,21,19)(20,22,24,23)(26,31)(27,33)(34,39)(35,40);(1,2,4,6)(3,5,7,8)(9,10)(11,13)(12,14)(15,16)(17,19,21,18)(20,23,24,22)(25,27)(26,32)(28,31)(29,35)(30,33)(34,37)(36,39)(38,40);(1,2)(3,5)(4,6)(7,8)(17,20)(18,22)(19,23)(21,24)(25,28,30,32)(26,33,31,27)(29,37,38,36)(34,35,39,40);(1,3,4,7)(2,5,6,8)(9,11,12,15)(10,13,14,16)(25,29)(26,34)(27,35)(28,36)(30,38)(31,39)(32,37)(33,40)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=fgi,d^2=f,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5706815);