Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2805613 |
| 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 | |
1548 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)(3,4)(5,7)(6,8)(9,10,14,13)(11,12,16,15)(17,18,21,24)(19,22,20,23)(25,26,30,31)(27,28,35,32)(29,37,38,33)(34,39,40,36);(1,3,5,8)(2,6,7,4)(9,11,14,16)(10,15,13,12)(26,31)(28,32)(29,38)(34,40);(9,12)(10,16)(11,13)(14,15)(17,19,21,20)(18,22,24,23)(25,27)(26,32)(28,31)(29,34)(30,35)(33,39)(36,37)(38,40);(1,2)(3,6)(4,8)(5,7)(9,12)(10,11)(13,16)(14,15)(17,18)(19,22)(20,23)(21,24)(25,28,30,32)(26,27,31,35)(29,39,38,36)(33,40,37,34);(1,4,5,6)(2,3,7,8)(9,13,14,10)(11,15,16,12)(17,20,21,19)(18,23,24,22)(25,29,30,38)(26,33,31,37)(27,34,35,40)(28,36,32,39)Polycyclic presentation:
2^9|a^2=i,b^2=fg,c^2=ghi,d^2=g,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2941341);