Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2735949 |
| 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 | |
1490 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)(3,4)(5,6)(7,8)(10,12)(14,15)(17,18,21,19)(20,23,24,22)(25,26)(27,28)(29,33)(30,31)(32,35)(34,36)(37,40)(38,39);(1,2,5,6)(3,4,7,8)(9,10,13,12)(11,14,16,15)(17,19,21,18)(20,23,24,22)(26,31)(28,35)(33,39)(36,37);(2,6)(3,7)(9,10,13,12)(11,15,16,14)(25,27)(26,28)(29,34)(30,32)(31,35)(33,36)(37,39)(38,40);(1,3)(2,4)(5,7)(6,8)(9,11)(10,14)(12,15)(13,16)(17,20,21,24)(18,22,19,23)(25,28)(26,32)(27,31)(29,37)(30,35)(33,34)(36,38)(39,40);(1,4)(2,7)(3,6)(5,8)(9,12,13,10)(11,14,16,15)(25,29,30,38)(26,33,31,39)(27,34,32,40)(28,36,35,37)Polycyclic presentation:
2^9|a^2=fgi,b^2=fi,c^2=gh,d^2=fgi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2654356);