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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1452 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,3)(4,6,8,7)(9,10,14,15)(11,13,12,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,31)(28,30)(29,36)(32,38)(33,35)(34,40)(37,39);(4,8)(6,7)(9,11,14,12)(10,13,15,16)(17,18)(19,20)(21,22)(23,24)(26,28)(27,33)(32,36)(34,37);(2,3)(6,7)(10,15)(11,12)(18,22)(19,23)(25,27,30,33)(26,31,28,35)(29,37,38,34)(32,40,36,39);(1,3,5,2)(4,7,8,6)(9,12,14,11)(10,13,15,16)(17,19,21,23)(18,20,22,24)(25,28,30,26)(27,31,33,35)(29,36,38,32)(34,39,37,40);(1,4,5,8)(2,6,3,7)(9,13)(10,11)(12,15)(14,16)(17,20)(18,23)(19,22)(21,24)(25,29)(26,32)(27,34)(28,36)(30,38)(31,39)(33,37)(35,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=fi,c^2=f,d^2=fg,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, 4980809);