Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| 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 | |
1360 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)(3,7)(4,6)(5,8)(9,10,13,14)(11,12,16,15)(17,18,21,24)(19,22,20,23)(25,26)(27,33)(28,30)(29,32)(31,35)(34,40)(36,37)(38,39);(1,3)(2,7)(4,5)(6,8)(9,10)(11,15)(12,16)(13,14)(17,19,21,20)(18,22,24,23)(26,28)(29,37)(31,33)(34,38);(2,4)(3,8)(10,14)(12,15)(17,19,21,20)(18,22,24,23)(25,27,30,35)(26,31,28,33)(29,38,37,34)(32,40,36,39);(1,4,6,2)(3,5,8,7)(9,11)(10,12)(13,16)(14,15)(17,18)(19,22)(20,23)(21,24)(25,28,30,26)(27,33,35,31)(29,36,37,32)(34,40,38,39);(1,5)(2,3)(4,8)(6,7)(9,12)(10,11)(13,15)(14,16)(17,20,21,19)(18,23,24,22)(25,29)(26,32)(27,34)(28,36)(30,37)(31,39)(33,40)(35,38)Polycyclic presentation:
2^9|a^2=gi,b^2=fhi,c^2=fgi,d^2=fh,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eh,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, 5116681);