Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #35769 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4395 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,4)(5,8)(6,7)(10,14)(15,17)(16,21)(22,23)(25,26)(27,32)(28,30)(29,34)(31,33)(35,40)(36,39)(37,38);(1,3)(2,5)(4,6)(7,8)(9,10,13,14)(11,17,18,15)(12,16,20,21)(19,23,24,22)(25,27,31,29)(26,32,33,34)(28,38,35,36)(30,37,40,39);(1,4)(2,8)(3,6)(5,7)(11,18)(12,20)(15,17)(16,21)(26,33)(27,29)(30,40)(36,38);(1,2,7,6)(3,4,8,5)(9,10)(11,17)(12,16)(13,14)(15,18)(19,23)(20,21)(22,24)(25,28,31,35)(26,30,33,40)(27,36,29,38)(32,39,34,37);(1,5,7,4)(2,8,6,3)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(25,29,31,27)(26,34,33,32)(28,36,35,38)(30,39,40,37);(1,6,7,2)(3,4,8,5)(9,12,13,20)(10,16,14,21)(11,19,18,24)(15,22,17,23)(25,30)(26,35)(27,37)(28,33)(29,39)(31,40)(32,36)(34,38)Polycyclic presentation:
2^9|b^2=gh,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8947939);