Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| 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 | |
1452 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)(3,8)(4,6)(5,7)(10,14)(12,15)(17,18,21,19)(20,23,24,22)(25,26)(27,28)(29,33)(30,31)(32,35)(34,36)(37,40)(38,39);(1,3,5,6)(2,4,7,8)(9,10,13,14)(11,12,16,15)(17,18,21,19)(20,22,24,23)(26,31)(28,32)(33,39)(36,37);(9,11)(10,15)(12,14)(13,16)(17,19,21,18)(20,23,24,22)(25,27)(26,32)(28,31)(29,34)(30,35)(33,37)(36,39)(38,40);(1,2)(3,4)(5,7)(6,8)(9,11)(10,12)(13,16)(14,15)(17,20)(18,22)(19,23)(21,24)(25,28,30,32)(26,27,31,35)(29,37,38,36)(33,40,39,34);(1,4)(2,6)(3,7)(5,8)(9,12)(10,16)(11,14)(13,15)(25,29,30,38)(26,33,31,39)(27,34,35,40)(28,36,32,37)Polycyclic presentation:
2^9|a^2=i,b^2=fgi,c^2=i,d^2=g,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4793574);