Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #6132643 |
| 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 | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1835 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)(3,4)(5,6)(7,8)(9,10,13,11)(12,15,16,14)(17,18)(19,22)(20,24)(21,23)(25,26,29,30)(27,31,33,37)(28,35,36,32)(34,40,39,38);(3,8)(4,7)(9,10,13,11)(12,15,16,14)(17,19,21,24)(18,22,23,20)(26,30)(27,33)(28,36)(38,40);(2,6)(3,8)(10,11)(14,15)(25,26)(27,31)(28,35)(29,30)(32,36)(33,37)(34,40)(38,39);(1,3,5,8)(2,7,6,4)(9,11,13,10)(12,15,16,14)(17,19)(18,20)(21,24)(22,23)(25,27)(26,31)(28,34)(29,33)(30,37)(32,38)(35,40)(36,39);(1,4,5,7)(2,8,6,3)(9,12,13,16)(10,14,11,15)(17,20)(18,19)(21,22)(23,24)(25,28,29,36)(26,32,30,35)(27,34,33,39)(31,38,37,40)Polycyclic presentation:
2^9|a^2=h,b^2=fgi,d^2=fg,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dfi,e^b=eg,d^c=dfg,e^c=ef62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6098363);