Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2805613 |
| 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 | |
1528 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)(3,8)(9,10)(11,12)(13,15)(14,16)(17,18,21,19)(20,23,24,22)(25,26)(27,33)(28,30)(29,36)(31,35)(32,37)(34,38)(39,40);(1,2,5,6)(3,7,8,4)(9,11,13,16)(10,14,15,12)(26,28)(29,37)(31,33)(34,40);(9,10)(11,12)(13,15)(14,16)(17,19,21,18)(20,23,24,22)(25,27)(26,31)(28,33)(29,34)(30,35)(32,38)(36,39)(37,40);(1,3)(2,7)(4,6)(5,8)(9,10)(11,14)(12,16)(13,15)(17,20)(18,22)(19,23)(21,24)(25,28,30,26)(27,33,35,31)(29,32,37,36)(34,38,40,39);(1,4,5,7)(2,8,6,3)(9,12,13,14)(10,11,15,16)(25,29,30,37)(26,32,28,36)(27,34,35,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=fhi,b^2=fg,c^2=fhi,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2800768);