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 | |
4289 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)(3,8)(4,11)(5,13)(6,7)(9,15)(10,12)(14,16)(17,18)(19,21)(20,23)(22,28)(24,26)(25,30)(27,31)(29,32)(33,34,39,37)(35,40,41,36)(38,45,46,42)(43,44,48,47);(1,3,6,11)(2,4,7,8)(5,10,15,14)(9,12,13,16)(17,19)(18,24)(20,21)(22,27)(23,26)(25,31)(28,29)(30,32)(33,35)(34,40)(36,37)(38,43)(39,41)(42,47)(44,45)(46,48);(2,7)(4,8)(9,13)(12,16)(18,20)(19,26)(25,28)(27,32)(33,36,39,40)(34,41,37,35)(38,44,46,47)(42,48,45,43);(1,4)(2,3)(5,12)(6,8)(7,11)(9,10)(13,14)(15,16)(17,20,23,18)(19,21,26,24)(22,28,30,25)(27,29,32,31)(33,36)(34,35)(37,41)(38,44)(39,40)(42,43)(45,48)(46,47);(1,3)(2,8)(4,7)(5,14)(6,11)(9,12)(10,15)(13,16)(17,21)(18,19)(20,26)(22,29)(23,24)(25,27)(28,32)(30,31)(33,37,39,34)(35,40,41,36)(38,42,46,45)(43,44,48,47);(1,5,6,15)(2,9,7,13)(3,10,11,14)(4,12,8,16)(17,22,23,30)(18,25,20,28)(19,27,26,32)(21,29,24,31)(33,38)(34,42)(35,43)(36,44)(37,45)(39,46)(40,47)(41,48)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=gh,d^2=ghi,e^2=gh,f^2=g,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, 10154550);