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 | |
1386 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,6)(3,7)(17,18,21,19)(20,23,24,22)(25,26)(27,30)(28,31)(29,34)(32,38)(33,36)(35,40)(37,39);(1,2,5,6)(3,8,7,4)(10,13)(14,16)(17,19,21,18)(20,23,24,22)(26,27)(31,33)(32,34)(39,40);(2,6)(3,7)(9,10,12,13)(11,14,15,16)(25,27,30,26)(28,31,36,33)(29,34,38,32)(35,39,37,40);(1,3,5,7)(2,4,6,8)(17,20,21,24)(18,22,19,23)(25,28)(26,31)(27,33)(29,37)(30,36)(32,40)(34,39)(35,38);(1,4)(2,3)(5,8)(6,7)(9,11,12,15)(10,14,13,16)(25,29)(26,32)(27,34)(28,35)(30,38)(31,39)(33,40)(36,37)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,c^2=fgh,d^2=i,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, 2654280);