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 | |
1534 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)(10,11)(12,16)(17,18,21,20)(19,22,24,23)(25,26)(27,35)(28,29)(30,31)(32,36)(33,34)(37,39)(38,40);(1,2,5,4)(3,6,7,8)(10,11)(14,15)(17,18,21,20)(19,23,24,22)(26,31)(27,36)(28,34)(38,39);(9,10,13,11)(12,14,16,15)(17,19,21,24)(18,22,20,23)(25,27)(26,32)(28,37)(29,38)(30,36)(31,35)(33,39)(34,40);(1,3)(2,6)(4,8)(5,7)(9,11,13,10)(12,15,16,14)(25,28)(26,33)(27,37)(29,31)(30,34)(32,39)(35,38)(36,40);(1,4,5,2)(3,8,7,6)(9,12,13,16)(10,14,11,15)(17,20,21,18)(19,23,24,22)(25,29,30,33)(26,34,31,28)(27,38,36,39)(32,40,35,37)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=fh,d^2=fg,e^2=ghi,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, 2634741);