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 | |
4071 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,5)(3,9,11,7)(4,12,14,8)(10,15,13,16)(17,18)(19,26)(20,23)(21,29)(22,25)(24,30)(27,31)(28,32)(33,34)(35,36)(37,40)(38,45)(39,41)(42,47)(43,48)(44,46);(1,3,6,11)(2,7,5,9)(4,13,14,10)(8,16,12,15)(19,20)(21,30)(23,26)(24,29)(33,35,40,39)(34,41,37,36)(38,43,47,46)(42,48,45,44);(2,5)(7,9)(8,12)(15,16)(17,19,22,20)(18,23,25,26)(21,27,30,28)(24,31,29,32)(33,36)(34,39)(35,37)(38,44)(40,41)(42,46)(43,45)(47,48);(1,4,6,14)(2,8,5,12)(3,10,11,13)(7,15,9,16)(17,20,22,19)(18,23,25,26)(21,28,30,27)(24,31,29,32)(33,37,40,34)(35,41,39,36)(38,45,47,42)(43,48,46,44);(1,3)(2,7)(4,10)(5,9)(6,11)(8,15)(12,16)(13,14)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,29)(26,32)(33,38)(34,42)(35,43)(36,44)(37,45)(39,46)(40,47)(41,48);(1,5,6,2)(3,9,11,7)(4,12,14,8)(10,16,13,15)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,31)(29,30)(33,39,40,35)(34,41,37,36)(38,46,47,43)(42,48,45,44)Polycyclic presentation:
2^9|a^2=hi,b^2=g,c^2=gh,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8109591);