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 | |
4232 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,11,13,7)(4,9,14,16)(5,10,15,12)(17,18)(19,24)(20,30)(21,25)(22,26)(23,27)(28,32)(29,31)(33,34)(35,39)(36,45)(37,44)(38,40)(41,46)(42,48)(43,47);(1,3,6,13)(2,7,8,11)(4,12,14,10)(5,9,15,16)(17,19,23,28)(18,24,27,32)(20,31,25,22)(21,26,30,29)(33,35,38,43)(34,39,40,47)(36,44,46,42)(37,41,48,45);(2,8)(5,15)(7,11)(9,16)(19,28)(22,31)(24,32)(26,29)(34,40)(35,43)(41,45)(42,44);(1,4,6,14)(2,9,8,16)(3,12,13,10)(5,7,15,11)(17,20)(18,21)(19,22)(23,25)(24,29)(26,32)(27,30)(28,31)(33,36)(34,41)(35,44)(37,47)(38,46)(39,48)(40,45)(42,43);(1,3)(2,7)(4,10)(5,9)(6,13)(8,11)(12,14)(15,16)(17,21)(18,25)(19,29)(20,27)(22,32)(23,30)(24,31)(26,28)(33,37,38,48)(34,42,40,44)(35,45,43,41)(36,39,46,47);(1,5)(2,10)(3,9)(4,7)(6,15)(8,12)(11,14)(13,16)(17,22)(18,26)(19,20)(21,32)(23,31)(24,30)(25,28)(27,29)(33,34)(35,39)(36,41)(37,42)(38,40)(43,47)(44,48)(45,46)Polycyclic presentation:
2^9|a^2=g,b^2=gi,d^2=g,e^2=gh,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, 8314318);