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 | #2619 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4262 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,17,19,12)(13,16,21,22)(18,23,24,20)(27,31)(28,34)(35,37)(36,39);(1,2,5,4)(3,7,8,6)(10,15)(12,17)(16,22)(20,23)(25,26,30,33)(27,34,31,28)(29,32,38,40)(35,39,37,36);(2,4)(6,7)(13,21)(16,22)(18,24)(20,23)(25,27,30,31)(26,28,33,34)(29,37,38,35)(32,39,40,36);(3,8)(6,7)(9,11,14,19)(10,12,15,17)(13,18,21,24)(16,20,22,23)(25,26)(27,28)(29,32)(30,33)(31,34)(35,36)(37,39)(38,40);(1,3)(2,6)(4,7)(5,8)(9,12)(10,11)(13,20)(14,17)(15,19)(16,18)(21,23)(22,24)(25,28)(26,31)(27,33)(29,36)(30,34)(32,37)(35,40)(38,39);(1,4,5,2)(3,7,8,6)(9,13,14,21)(10,16,15,22)(11,18,19,24)(12,20,17,23)(25,29,30,38)(26,32,33,40)(27,35,31,37)(28,36,34,39)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=h,d^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8281534);