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 | |
3977 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
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)(17,18,22,25)(19,21,27,23)(20,24,28,31)(26,29,32,30)(33,34)(35,46)(36,42)(37,47)(38,43)(39,44)(40,45)(41,48);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(20,28)(24,31)(26,32)(29,30)(33,35,40,48)(34,41,45,46)(36,38,37,44)(39,47,43,42);(2,6)(7,9)(8,12)(14,15)(17,19,22,27)(18,23,25,21)(20,26,28,32)(24,30,31,29)(33,36,40,37)(34,42,45,47)(35,38,48,44)(39,46,43,41);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,19)(18,21)(20,26)(22,27)(23,25)(24,29)(28,32)(30,31)(33,37,40,36)(34,42,45,47)(35,38,48,44)(39,41,43,46);(1,3,5,11)(2,7,6,9)(4,10,13,16)(8,14,12,15)(17,20,22,28)(18,24,25,31)(19,26,27,32)(21,29,23,30)(33,38)(34,43)(35,37)(36,48)(39,45)(40,44)(41,42)(46,47);(1,4,5,13)(2,8,6,12)(3,10,11,16)(7,14,9,15)(17,21)(18,19)(20,29)(22,23)(24,26)(25,27)(28,30)(31,32)(33,39,40,43)(34,44,45,38)(35,47,48,42)(36,46,37,41)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=gh,d^2=gi,e^2=h,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, 10154176);