Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1561 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,3)(4,6,8,7)(9,10,13,14)(11,15,16,12)(17,18,21,19)(20,23,24,22)(25,26)(27,31)(28,32)(29,35)(30,33)(34,39)(36,37)(38,40);(2,3)(6,7)(10,14)(11,16)(25,27,30,35)(26,31,33,29)(28,36,38,34)(32,37,40,39);(4,8)(6,7)(9,11,13,16)(10,15,14,12)(17,19,21,18)(20,23,24,22)(25,27)(26,29)(28,34)(30,35)(31,33)(32,37)(36,38)(39,40);(1,3,5,2)(4,6,8,7)(9,10)(11,15)(12,16)(13,14)(17,20,21,24)(18,22,19,23)(25,28,30,38)(26,32,33,40)(27,34,35,36)(29,37,31,39);(1,4,5,8)(2,6,3,7)(9,12,13,15)(10,11,14,16)(25,29)(26,27)(28,37)(30,31)(32,34)(33,35)(36,40)(38,39)Polycyclic presentation:
2^9|a^2=fghi,b^2=g,c^2=i,d^2=fgi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,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, 3454262);