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 | |
4144 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)(3,11)(4,9)(5,10)(6,8)(7,14)(12,16)(13,15)(17,18)(19,24)(20,30)(21,25)(22,26)(23,27)(28,32)(29,31)(33,34,39,40)(35,41,45,48)(36,42,44,38)(37,43,46,47);(1,3,6,14)(2,7,8,11)(4,12,15,10)(5,13,16,9)(17,19,23,28)(18,24,27,32)(20,31,25,22)(21,26,30,29)(34,40)(36,44)(37,46)(41,48);(2,8)(5,16)(7,11)(9,13)(19,28)(22,31)(24,32)(26,29)(33,35,39,45)(34,41,40,48)(36,37,44,46)(38,47,42,43);(1,4,6,15)(2,9,8,13)(3,12,14,10)(5,11,16,7)(17,20)(18,21)(19,22)(23,25)(24,29)(26,32)(27,30)(28,31)(33,36)(34,42)(35,37)(38,40)(39,44)(41,43)(45,46)(47,48);(1,3)(2,7)(4,10)(5,13)(6,14)(8,11)(9,16)(12,15)(17,21)(18,25)(19,29)(20,27)(22,32)(23,30)(24,31)(26,28)(33,37)(34,43)(35,44)(36,45)(38,41)(39,46)(40,47)(42,48);(1,5,6,16)(2,10,8,12)(3,13,14,9)(4,11,15,7)(17,22)(18,26)(19,20)(21,32)(23,31)(24,30)(25,28)(27,29)(33,38,39,42)(34,36,40,44)(35,43,45,47)(37,41,46,48)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=gh,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8314274);