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 | |
4004 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)(3,6)(4,11)(5,7)(8,12)(9,15)(10,13)(14,16)(17,18)(19,20)(21,29)(22,23)(24,31)(25,26)(27,30)(28,32)(33,34)(35,41)(36,38)(37,40)(39,43)(42,47)(44,45)(46,48);(1,3,5,10)(2,6,7,13)(4,9,12,16)(8,14,11,15)(17,18)(19,25)(20,26)(21,24)(22,23)(27,30)(28,32)(29,31)(33,35,39,37)(34,40,43,41)(36,46,47,44)(38,45,42,48);(2,7)(6,13)(8,11)(14,15)(18,23)(19,26)(24,29)(27,32)(33,35,39,37)(34,41,43,40)(36,44,47,46)(38,45,42,48);(1,2)(3,6)(4,8)(5,7)(9,14)(10,13)(11,12)(15,16)(17,19,22,26)(18,20,23,25)(21,27,31,32)(24,28,29,30)(33,36)(34,42)(35,44)(37,46)(38,43)(39,47)(40,45)(41,48);(1,3)(2,6)(4,9)(5,10)(7,13)(8,14)(11,15)(12,16)(17,20)(18,19)(21,30)(22,25)(23,26)(24,32)(27,29)(28,31)(33,37,39,35)(34,40,43,41)(36,44,47,46)(38,45,42,48);(1,4,5,12)(2,8,7,11)(3,9,10,16)(6,14,13,15)(17,21,22,31)(18,24,23,29)(19,27,26,32)(20,28,25,30)(33,38,39,42)(34,36,43,47)(35,45,37,48)(40,46,41,44)Polycyclic presentation:
2^9|b^2=ghi,c^2=g,d^2=i,e^2=g,f^2=gh,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, 10154653);