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 | |
4046 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)(3,7)(4,12)(5,6)(8,13)(9,10)(11,15)(14,16)(17,18)(19,27)(20,29)(21,26)(22,30)(23,25)(24,28)(31,32)(33,34,39,36)(35,40,45,43)(37,41,44,46)(38,42,48,47);(1,2)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,19,23,28)(18,24,25,27)(20,21,30,31)(22,26,29,32)(33,35,39,45)(34,40,36,43)(37,38,44,48)(41,42,46,47);(2,6)(3,10)(8,12)(11,16)(18,25)(19,28)(20,30)(26,32)(33,34,39,36)(35,43,45,40)(37,46,44,41)(38,42,48,47);(1,2)(3,9)(4,12)(5,6)(7,10)(8,13)(11,14)(15,16)(17,20)(18,22)(19,21)(23,30)(24,26)(25,29)(27,32)(28,31)(33,36,39,34)(35,43,45,40)(37,41,44,46)(38,42,48,47);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,15)(13,16)(17,21)(18,26)(19,20)(22,24)(23,31)(25,32)(27,29)(28,30)(33,37,39,44)(34,41,36,46)(35,38,45,48)(40,42,43,47);(1,4,5,13)(2,8,6,12)(3,11,10,16)(7,14,9,15)(17,22)(18,20)(19,26)(21,24)(23,29)(25,30)(27,31)(28,32)(33,38)(34,42)(35,44)(36,47)(37,45)(39,48)(40,46)(41,43)Polycyclic presentation:
2^9|a^2=hi,b^2=ghi,c^2=hi,d^2=hi,e^2=hi,f^2=gh,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, 9575563);