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 | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1875 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,6,8)(3,4,5,7)(9,10)(11,16)(12,14)(13,15)(17,18,21,23)(19,20,24,22)(25,26,29,32)(27,33,34,30)(28,36,37,31)(35,38,40,39);(1,3)(2,7)(4,8)(5,6)(9,10,13,15)(11,14,12,16)(17,18)(19,22)(20,24)(21,23)(27,34)(28,37)(30,33)(31,36);(2,8)(3,5)(9,11,13,12)(10,14,15,16)(17,19,21,24)(18,22,23,20)(25,27)(26,30)(28,35)(29,34)(31,38)(32,33)(36,39)(37,40);(1,4)(2,3)(5,8)(6,7)(9,12,13,11)(10,14,15,16)(17,20)(18,19)(21,22)(23,24)(25,26)(27,33)(28,31)(29,32)(30,34)(35,39)(36,37)(38,40);(1,5,6,3)(2,4,8,7)(9,10)(11,16)(12,14)(13,15)(17,19)(18,20)(21,24)(22,23)(25,28)(26,31)(27,35)(29,37)(30,38)(32,36)(33,39)(34,40)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=gh,d^2=gi,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, 5025343);