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 | #56401 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4376 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
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)(25,26,32,36)(27,37,39,33)(28,30,29,34)(31,35,40,38);(1,2)(3,7)(4,5)(6,8)(10,15)(12,17)(16,22)(20,23)(25,27,32,39)(26,33,36,37)(28,38,29,35)(30,40,34,31);(2,5)(6,7)(13,21)(16,22)(18,24)(20,23)(25,28,32,29)(26,34,36,30)(27,38,39,35)(31,37,40,33);(9,11)(10,12)(13,18)(14,19)(15,17)(16,20)(21,24)(22,23)(25,29,32,28)(26,30,36,34)(27,38,39,35)(31,37,40,33);(1,3,4,8)(2,6,5,7)(9,12)(10,11)(13,20)(14,17)(15,19)(16,18)(21,23)(22,24)(25,30)(26,28)(27,31)(29,36)(32,34)(33,35)(37,38)(39,40);(1,2)(3,6)(4,5)(7,8)(9,13,14,21)(10,16,15,22)(11,18,19,24)(12,20,17,23)(25,31)(26,35)(27,30)(28,33)(29,37)(32,40)(34,39)(36,38)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=h,d^2=h,e^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8122642);