Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2031997 |
| automorphism | ? |
| 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 | |
1867 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,5)(3,7,8,6)(9,10,13,14)(11,15,12,16)(17,18,21,22)(19,20,23,24)(26,27)(28,35)(32,34)(36,40);(2,5)(6,7)(10,14)(15,16)(18,22)(19,23)(25,26,30,27)(28,33,35,31)(29,34,37,32)(36,38,40,39);(9,11,13,12)(10,15,14,16)(17,19)(18,20)(21,23)(22,24)(25,27,30,26)(28,33,35,31)(29,34,37,32)(36,39,40,38);(1,3,4,8)(2,6,5,7)(9,12,13,11)(10,16,14,15)(17,20)(18,19)(21,24)(22,23)(25,28,30,35)(26,31,27,33)(29,36,37,40)(32,38,34,39);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,20)(21,22)(23,24)(25,29,30,37)(26,32,27,34)(28,36,35,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=fg,b^2=g,c^2=fgi,d^2=fh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1957790);