Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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 | |
1522 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10,13,14)(11,12,16,15)(18,20)(22,23)(25,26,30,31)(27,29,35,32)(28,36,39,33)(34,38,37,40);(4,8)(6,7)(9,11,13,16)(10,12,14,15)(18,20)(22,23)(26,31)(29,32)(33,36)(38,40);(2,3)(6,7)(10,14)(12,15)(17,18,21,20)(19,23,24,22)(25,27,30,35)(26,32,31,29)(28,37,39,34)(33,40,36,38);(1,3,5,2)(4,7,8,6)(9,12,13,15)(10,11,14,16)(17,19,21,24)(18,22,20,23)(25,28)(26,33)(27,34)(29,40)(30,39)(31,36)(32,38)(35,37);(1,4)(2,6)(3,7)(5,8)(9,11)(10,15)(12,14)(13,16)(17,20,21,18)(19,23,24,22)(25,29,30,32)(26,27,31,35)(28,38,39,40)(33,37,36,34)Polycyclic presentation:
2^9|a^2=gh,b^2=fi,c^2=fh,d^2=fg,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,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, 4086935);