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 | #47675 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4283 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)(3,8)(4,6)(5,7)(9,10,14,19)(11,16,22,15)(12,17,18,13)(20,24,23,21)(25,26)(27,33)(28,36)(29,30)(31,34)(32,37)(35,39)(38,40);(1,3,5,6)(2,4,7,8)(9,11,14,22)(10,15,19,16)(12,23,18,20)(13,21,17,24)(27,34)(28,37)(31,33)(32,36);(1,4)(2,6)(3,7)(5,8)(9,11,14,22)(10,15,19,16)(12,23,18,20)(13,21,17,24)(25,27,30,34)(26,31,29,33)(28,38,37,35)(32,40,36,39);(1,2)(3,8)(4,6)(5,7)(9,11)(10,16)(12,20)(13,21)(14,22)(15,19)(17,24)(18,23)(25,28,30,37)(26,32,29,36)(27,35,34,38)(31,39,33,40);(1,2)(3,4)(5,7)(6,8)(9,12)(10,17)(11,20)(13,19)(14,18)(15,21)(16,24)(22,23)(25,27)(26,33)(28,35)(29,31)(30,34)(32,40)(36,39)(37,38);(1,4)(2,6)(3,7)(5,8)(9,13,14,17)(10,18,19,12)(11,21,22,24)(15,20,16,23)(25,29,30,26)(27,33,34,31)(28,36,37,32)(35,40,38,39)Polycyclic presentation:
2^9|a^2=hi,b^2=g,c^2=h,d^2=i,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9947694);