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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4901 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(9,10)(11,14)(12,15)(13,16)(17,18,22,23)(19,27,30,24)(20,25,31,35)(21,33,34,26)(28,39,41,36)(29,37,43,40)(32,45,44,38)(42,46,48,47);(1,2)(3,7)(4,6)(5,8)(9,11)(10,13)(12,16)(14,15)(18,23)(20,31)(24,27)(26,33)(28,41)(32,44)(37,40)(42,48);(1,3)(2,5)(4,8)(6,7)(17,18)(19,27)(20,25)(21,26)(22,23)(24,30)(28,39)(29,40)(31,35)(32,38)(33,34)(36,41)(37,43)(42,47)(44,45)(46,48);(2,6)(5,7)(9,11,12,16)(10,13,15,14)(17,19,22,30)(18,24,23,27)(20,28,31,41)(21,29,34,43)(25,36,35,39)(26,37,33,40)(32,42,44,48)(38,46,45,47);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,16)(13,15)(17,20)(18,25)(19,28)(21,32)(22,31)(23,35)(24,36)(26,38)(27,39)(29,42)(30,41)(33,45)(34,44)(37,46)(40,47)(43,48);(1,3)(2,7)(4,8)(5,6)(9,10)(11,13)(12,15)(14,16)(17,21,22,34)(18,26,23,33)(19,29,30,43)(20,32,31,44)(24,37,27,40)(25,38,35,45)(28,42,41,48)(36,46,39,47)Polycyclic presentation:
2^9|a^2=h,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10126985);