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 | |
4910 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,7)(3,4,8,6)(18,19)(20,30)(22,36)(25,28)(31,33)(34,37)(39,43)(47,48);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16)(17,18,23,19)(20,27,30,24)(21,28,35,25)(22,26,36,29)(31,38,33,40)(32,43,44,39)(34,42,37,41)(45,47,46,48);(9,11)(10,13)(12,16)(14,15)(17,19,23,18)(20,24,30,27)(21,28,35,25)(22,29,36,26)(31,38,33,40)(32,39,44,43)(34,42,37,41)(45,47,46,48);(2,7)(3,8)(10,14)(13,15)(17,20,23,30)(18,24,19,27)(21,33,35,31)(22,32,36,44)(25,40,28,38)(26,39,29,43)(34,46,37,45)(41,48,42,47);(1,3)(2,4)(5,8)(6,7)(9,10)(11,13)(12,14)(15,16)(17,21)(18,25)(19,28)(20,31)(22,37)(23,35)(24,38)(26,42)(27,40)(29,41)(30,33)(32,46)(34,36)(39,48)(43,47)(44,45);(1,4)(2,8)(3,7)(5,6)(9,11)(10,15)(12,16)(13,14)(17,22)(18,26)(19,29)(20,32)(21,34)(23,36)(24,39)(25,41)(27,43)(28,42)(30,44)(31,45)(33,46)(35,37)(38,47)(40,48)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=h,d^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, 10125853);