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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1711 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)(9,10,13,12)(11,15,16,14)(18,22)(20,24)(26,30)(27,33)(28,35)(38,40);(2,3)(6,7)(17,18,21,22)(19,20,23,24)(26,30)(27,33)(32,37)(34,39);(1,2,5,3)(4,7,8,6)(9,10,13,12)(11,14,16,15)(17,19,21,23)(18,20,22,24)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(1,3,5,2)(4,6,8,7)(9,11,13,16)(10,14,12,15)(17,20,21,24)(18,19,22,23)(25,27,29,33)(26,31,30,36)(28,34,35,39)(32,38,37,40);(1,4)(2,6)(3,7)(5,8)(9,12,13,10)(11,15,16,14)(17,19)(18,20)(21,23)(22,24)(25,28,29,35)(26,32,30,37)(27,34,33,39)(31,38,36,40)Polycyclic presentation:
2^9|a^2=fgi,b^2=fgh,c^2=hi,d^2=ghi,e^2=fghi,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, 3764060);