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 | |
4542 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(9,10,13,11)(12,15,16,14)(18,23)(19,27)(21,34)(25,31)(29,39)(32,45)(37,42)(46,48);(1,2)(3,4)(5,6)(7,8)(10,11)(14,15)(17,18,22,23)(19,24,27,28)(20,31,33,25)(21,26,34,35)(29,41,39,36)(30,37,40,42)(32,44,45,38)(43,48,47,46);(2,6)(3,7)(9,11,13,10)(12,15,16,14)(17,18)(19,28)(20,25)(21,26)(22,23)(24,27)(29,41)(30,42)(31,33)(32,38)(34,35)(36,39)(37,40)(43,48)(44,45)(46,47);(3,7)(4,8)(9,12)(10,14)(11,15)(13,16)(17,19)(18,24)(20,29)(21,30)(22,27)(23,28)(25,36)(26,37)(31,41)(32,43)(33,39)(34,40)(35,42)(38,46)(44,48)(45,47);(1,3)(2,4)(5,7)(6,8)(17,20,22,33)(18,25,23,31)(19,29,27,39)(21,32,34,45)(24,36,28,41)(26,38,35,44)(30,43,40,47)(37,46,42,48);(1,4)(2,3)(5,8)(6,7)(17,21)(18,26)(19,30)(20,32)(22,34)(23,35)(24,37)(25,38)(27,40)(28,42)(29,43)(31,44)(33,45)(36,46)(39,47)(41,48)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=gh,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,e^d=egi,f^d=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9532611);