Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| 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 | |
1572 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)(3,8)(4,5)(6,7)(9,10,13,12)(11,15,16,14)(19,23)(22,24)(25,26,30,28)(27,33,34,31)(29,36,37,32)(35,38,40,39);(1,3)(2,5)(4,6)(7,8)(9,11,13,16)(10,14,12,15)(18,21)(22,24)(27,34)(29,37)(31,33)(32,36);(3,4)(5,8)(9,11,13,16)(10,15,12,14)(17,18,20,21)(19,22,23,24)(25,27)(26,31)(28,33)(29,35)(30,34)(32,38)(36,39)(37,40);(1,4,6,3)(2,5,7,8)(9,11)(10,14)(12,15)(13,16)(25,28,30,26)(27,31,34,33)(29,36,37,32)(35,38,40,39);(1,5)(2,3)(4,7)(6,8)(9,12,13,10)(11,15,16,14)(17,19,20,23)(18,22,21,24)(25,29)(26,32)(27,35)(28,36)(30,37)(31,38)(33,39)(34,40)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=fgh,d^2=fgi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5025333);