Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1758 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,7)(3,8,4,6)(9,10)(11,12)(13,15)(14,16)(17,18,22,21)(19,24,20,23)(25,26,29,32)(27,33,35,30)(28,36,37,31)(34,38,40,39);(1,3,5,4)(2,6,7,8)(9,11)(10,14)(12,15)(13,16)(17,19,22,20)(18,23,21,24)(28,37)(31,36)(34,40)(38,39);(3,4)(6,8)(11,16)(12,14)(17,19,22,20)(18,24,21,23)(25,27,29,35)(26,30,32,33)(28,34,37,40)(31,38,36,39);(1,2)(3,6)(4,8)(5,7)(9,10)(11,14)(12,16)(13,15)(17,20,22,19)(18,23,21,24)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,37)(38,40);(1,4,5,3)(2,6,7,8)(9,12,13,14)(10,11,15,16)(17,21,22,18)(19,24,20,23)(25,28)(26,31)(27,34)(29,37)(30,38)(32,36)(33,39)(35,40)Polycyclic presentation:
2^9|a^2=fhi,b^2=fghi,c^2=fgh,d^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,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, 4105816);