Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
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 | |
1480 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,4)(3,7)(9,10,13,12)(11,14,15,16)(17,18,22,20)(19,23,21,24)(25,26)(27,28)(29,37)(30,31)(32,34)(33,40)(35,39)(36,38);(3,7)(6,8)(11,15)(14,16)(17,19,22,21)(18,23,20,24)(26,31)(27,34)(33,37)(35,38);(1,2,5,4)(3,6,7,8)(9,11,13,15)(10,14,12,16)(17,20,22,18)(19,23,21,24)(25,27)(26,32)(28,31)(29,38)(30,34)(33,36)(35,40)(37,39);(1,3,5,7)(2,6,4,8)(9,12,13,10)(11,16,15,14)(17,21,22,19)(18,23,20,24)(25,28,30,32)(26,27,31,34)(29,39,40,36)(33,38,37,35);(1,4,5,2)(3,8,7,6)(25,29)(26,33)(27,35)(28,36)(30,40)(31,37)(32,39)(34,38)Polycyclic presentation:
2^9|a^2=fh,b^2=fgi,c^2=gh,d^2=fgh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,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, 3286980);