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 | #59807 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4476 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,5)(3,7,8,6)(9,10,14,15)(11,19,21,16)(12,20,22,17)(13,18,24,23)(25,26,31,35)(27,34,29,32)(28,33,38,30)(36,40,39,37);(2,5)(6,7)(10,15)(12,22)(16,19)(18,23)(27,29)(32,34)(36,39)(37,40);(1,2)(3,6)(4,5)(7,8)(10,15)(13,24)(16,19)(17,20)(25,27,31,29)(26,32,35,34)(28,36,38,39)(30,40,33,37);(1,2)(3,6)(4,5)(7,8)(9,11,14,21)(10,16,15,19)(12,23,22,18)(13,20,24,17)(25,28)(26,33)(27,36)(29,39)(30,35)(31,38)(32,37)(34,40);(1,3)(2,6)(4,8)(5,7)(9,12)(10,17)(11,18)(13,19)(14,22)(15,20)(16,24)(21,23)(25,29,31,27)(26,34,35,32)(28,36,38,39)(30,40,33,37);(9,13,14,24)(10,18,15,23)(11,20,21,17)(12,19,22,16)(25,30,31,33)(26,28,35,38)(27,37,29,40)(32,39,34,36)Polycyclic presentation:
2^9|a^2=h,c^2=gh,d^2=hi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=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, 8309535);