Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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 | |
1813 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(6,8)(11,12)(14,15)(17,18,21,22)(19,20,23,24)(25,26,29,30)(27,31,33,34)(28,32,36,38)(35,39,37,40);(9,10,13,16)(11,14,12,15)(17,19,21,23)(18,20,22,24)(26,30)(27,33)(28,36)(39,40);(1,2,5,4)(3,6,7,8)(9,11,13,12)(10,14,16,15)(17,20)(18,19)(21,24)(22,23)(25,26)(27,34)(28,32)(29,30)(31,33)(35,40)(36,38)(37,39);(1,3)(2,6)(4,8)(5,7)(9,10)(11,14)(12,15)(13,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,31)(28,37)(29,33)(30,34)(32,40)(35,36)(38,39);(1,4,5,2)(3,8,7,6)(9,12,13,11)(10,15,16,14)(17,18)(19,20)(21,22)(23,24)(25,28,29,36)(26,32,30,38)(27,35,33,37)(31,39,34,40)Polycyclic presentation:
2^9|a^2=h,b^2=fg,c^2=ghi,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3163835);