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 | #36722 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4278 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10)(11,16)(12,22)(13,17)(14,23)(15,18)(19,20)(21,24)(26,30)(28,35)(31,36)(34,39);(2,6)(3,7)(9,11,15,21)(10,16,18,24)(12,17,23,19)(13,14,20,22)(27,33)(31,36)(34,39)(38,40);(1,3)(2,4)(5,7)(6,8)(9,12,15,23)(10,14,18,22)(11,19,21,17)(13,24,20,16)(25,26,29,30)(27,31,33,36)(28,32,35,37)(34,38,39,40);(1,4)(2,3)(5,8)(6,7)(9,12)(10,14)(11,17)(13,24)(15,23)(16,20)(18,22)(19,21);(9,13,15,20)(10,17,18,19)(11,14,21,22)(12,16,23,24)(25,27)(26,31)(28,34)(29,33)(30,36)(32,38)(35,39)(37,40);(9,14)(10,12)(11,20)(13,21)(15,22)(16,17)(18,23)(19,24)(25,28)(26,32)(27,34)(29,35)(30,37)(31,38)(33,39)(36,40)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=h,e^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8961116);