Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4379698 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 | |
2138 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)(6,8)(9,10,15,18)(11,13,12,16)(14,17,22,24)(19,21,20,23)(25,26)(27,32)(28,30)(29,36)(31,35)(33,37)(34,38)(39,40);(9,11,15,12)(10,16,18,13)(14,19,22,20)(17,23,24,21)(25,27,30,35)(26,31,28,32)(29,34,37,40)(33,38,36,39);(5,6)(7,8)(9,12,15,11)(10,16,18,13)(14,19,22,20)(17,21,24,23)(25,28,30,26)(27,31,35,32)(29,33,37,36)(34,39,40,38);(1,2,3,4)(9,13,15,16)(10,11,18,12)(14,21,22,23)(17,19,24,20)(25,27)(26,32)(28,31)(29,34)(30,35)(33,39)(36,38)(37,40);(9,14)(10,17)(11,19)(12,20)(13,21)(15,22)(16,23)(18,24)(25,29,30,37)(26,33,28,36)(27,34,35,40)(31,38,32,39)Polycyclic presentation:
2^9|a^2=fh,b^2=f,c^2=f,d^2=i,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4296956);