Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4379698 |
| 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 | |
1532 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)(6,7)(9,10,12,14)(11,15,16,13)(17,18,21,24)(19,22,20,23)(25,26)(27,33)(28,35)(29,32)(30,31)(34,40)(36,37)(38,39);(9,10,12,14)(11,13,16,15)(26,31)(28,33)(32,39)(36,40);(1,2,5,3)(4,6,8,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,22)(20,23)(21,24)(25,27)(26,28)(29,37)(30,35)(31,33)(32,40)(34,38)(36,39);(1,3,5,2)(4,7,8,6)(17,19,21,20)(18,22,24,23)(25,28)(26,27)(29,36)(30,33)(31,35)(32,34)(37,39)(38,40);(1,4,5,8)(2,6,3,7)(9,11,12,16)(10,13,14,15)(17,20,21,19)(18,23,24,22)(25,29)(26,32)(27,34)(28,36)(30,38)(31,39)(33,40)(35,37)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=fi,d^2=fgh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4295588);