Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
1428 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)(10,13)(14,16)(17,18,21,24)(19,22,20,23)(25,26,30,31)(27,28,32,35)(29,33,39,40)(34,36,37,38);(1,3,6,5)(2,7,4,8)(11,15)(14,16)(26,31)(28,35)(33,40)(36,38);(1,2,6,4)(3,8,5,7)(9,10)(11,14)(12,13)(15,16)(17,19,21,20)(18,22,24,23)(25,27,30,32)(26,28,31,35)(29,37,39,34)(33,38,40,36);(1,4,6,2)(3,7,5,8)(9,11,12,15)(10,14,13,16)(17,18)(19,22)(20,23)(21,24)(25,28)(26,32)(27,31)(29,38)(30,35)(33,34)(36,39)(37,40);(1,5,6,3)(2,7,4,8)(9,10)(11,14)(12,13)(15,16)(17,20,21,19)(18,23,24,22)(25,29,30,39)(26,33,31,40)(27,34,32,37)(28,36,35,38)Polycyclic presentation:
2^9|a^2=fhi,b^2=h,c^2=fi,d^2=g,e^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3604734);