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 | |
1428 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)(3,6)(4,7)(5,8)(10,14)(11,15)(25,26,29,30)(27,31,35,33)(28,36,37,32)(34,40,38,39);(1,3)(2,5)(4,8)(6,7)(9,10,13,14)(11,12,15,16)(17,18,21,20)(19,23,24,22)(26,30)(28,37)(31,33)(34,38);(3,8)(5,6)(10,14)(12,16)(17,19,21,24)(18,22,20,23)(25,26)(27,33)(28,36)(29,30)(31,35)(32,37)(34,39)(38,40);(1,2)(3,5)(4,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,18,21,20)(19,22,24,23)(25,27)(26,31)(28,38)(29,35)(30,33)(32,40)(34,37)(36,39);(1,3)(2,6)(4,8)(5,7)(9,12)(10,15)(11,14)(13,16)(17,20,21,18)(19,23,24,22)(25,28)(26,32)(27,34)(29,37)(30,36)(31,39)(33,40)(35,38)Polycyclic presentation:
2^9|a^2=g,b^2=fhi,c^2=fh,d^2=fh,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, 4095895);