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 | |
1456 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)(3,4,8,6)(11,15)(14,16)(17,18,21,19)(20,23,24,22)(25,26)(27,28)(29,33)(30,31)(32,35)(34,36)(37,40)(38,39);(1,3)(2,6)(4,7)(5,8)(10,13)(14,16)(17,19,21,18)(20,23,24,22)(26,31)(28,35)(33,39)(36,40);(2,7)(3,8)(9,10,12,13)(11,14,15,16)(25,27)(26,28)(29,37)(30,32)(31,35)(33,40)(34,38)(36,39);(1,4)(2,3)(5,6)(7,8)(17,20)(18,22)(19,23)(21,24)(25,28)(26,32)(27,31)(29,36)(30,35)(33,37)(34,39)(38,40);(1,2)(3,4)(5,7)(6,8)(9,11,12,15)(10,14,13,16)(25,29)(26,33)(27,34)(28,36)(30,38)(31,39)(32,37)(35,40)Polycyclic presentation:
2^9|a^2=fgi,b^2=fi,c^2=gh,e^2=gh,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, 4983726);