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) = C26 |
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 | |
1550 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,3)(4,8)(19,23)(21,24)(25,26,29,30)(27,31,33,38)(28,32,37,36)(34,39,35,40);(1,2,5,3)(4,6,8,7)(10,13)(14,16)(17,18,20,22)(19,21,23,24)(26,30)(27,33)(32,36)(34,35);(1,2,5,3)(4,7,8,6)(9,10)(11,14)(12,13)(15,16)(17,18)(19,21)(20,22)(23,24)(25,27)(26,31)(28,35)(29,33)(30,38)(32,40)(34,37)(36,39);(1,3,5,2)(4,6,8,7)(9,11,12,15)(10,14,13,16)(17,19)(18,21)(20,23)(22,24)(25,26)(27,31)(28,36)(29,30)(32,37)(33,38)(34,40)(35,39);(1,4,5,8)(2,6,3,7)(25,28,29,37)(26,32,30,36)(27,34,33,35)(31,39,38,40)Polycyclic presentation:
2^9|a^2=fg,b^2=fgi,c^2=g,d^2=hi,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, 3408476);