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 | |
1784 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)(9,10)(11,14)(12,16)(13,15)(17,18,21,22)(19,20,23,24)(25,26)(27,30)(28,31)(29,34)(32,38)(33,36)(35,40)(37,39);(2,3)(6,7)(9,11,13,16)(10,14,15,12)(17,19,21,23)(18,20,22,24)(26,27)(31,33)(32,34)(39,40);(1,2,5,3)(4,6,8,7)(9,12)(10,11)(13,14)(15,16)(17,20)(18,19)(21,24)(22,23)(25,27,30,26)(28,33,36,31)(29,32,38,34)(35,39,37,40);(1,3,5,2)(4,7,8,6)(9,11)(10,12)(13,16)(14,15)(17,19)(18,20)(21,23)(22,24)(25,28,30,36)(26,31,27,33)(29,37,38,35)(32,40,34,39);(1,4)(2,6)(3,7)(5,8)(9,12)(10,11)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,29,30,38)(26,32,27,34)(28,35,36,37)(31,39,33,40)Polycyclic presentation:
2^9|a^2=fgi,b^2=fhi,c^2=fghi,d^2=fghi,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, 3494312);