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 | |
4087 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,7,11,9)(4,12,13,8)(10,15,16,14)(17,18,22,26)(19,27,29,23)(20,24,31,28)(21,25,32,30)(33,34,38,40)(35,36,43,39)(37,45,47,41)(42,48,46,44);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,19,22,29)(18,23,26,27)(20,25,31,30)(21,28,32,24)(33,35,38,43)(34,39,40,36)(37,42,47,46)(41,48,45,44);(2,6)(7,9)(8,12)(14,15)(34,40)(36,39)(41,45)(44,48);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,19)(18,23)(20,30)(21,28)(22,29)(24,32)(25,31)(26,27)(33,36,38,39)(34,35,40,43)(37,44,47,48)(41,42,45,46);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,15)(13,16)(17,20)(18,24)(19,25)(21,23)(22,31)(26,28)(27,32)(29,30)(33,35)(34,36)(37,46)(38,43)(39,40)(41,48)(42,47)(44,45);(1,4,5,13)(2,8,6,12)(3,10,11,16)(7,14,9,15)(17,21)(18,25)(19,28)(20,27)(22,32)(23,31)(24,29)(26,30)(33,37,38,47)(34,41,40,45)(35,42,43,46)(36,44,39,48)Polycyclic presentation:
2^9|a^2=g,b^2=gi,d^2=hi,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, 10154845);