Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2601293 |
| 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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1479 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(6,8)(9,10,14,12)(11,15,13,16)(18,22)(20,23)(25,26,29,30)(27,31,33,38)(28,35,37,32)(34,40,36,39);(1,2,5,4)(3,6,7,8)(9,11,14,13)(10,15,12,16)(18,22)(19,24)(26,30)(27,33)(32,35)(34,36);(9,11,14,13)(10,16,12,15)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,32)(29,30)(33,38)(34,39)(35,37)(36,40);(1,3)(2,6)(4,8)(5,7)(9,12,14,10)(11,15,13,16)(17,19,21,24)(18,23,22,20)(25,27)(26,31)(28,36)(29,33)(30,38)(32,40)(34,37)(35,39);(1,4,5,2)(3,8,7,6)(9,13,14,11)(10,15,12,16)(17,20,21,23)(18,24,22,19)(25,28)(26,32)(27,34)(29,37)(30,35)(31,39)(33,36)(38,40)Polycyclic presentation:
2^9|a^2=fgh,b^2=ghi,c^2=gh,d^2=fg,e^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2618562);