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 | |
4221 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,5)(3,9,10,7)(4,12,14,8)(11,15,13,16)(17,18)(19,21)(20,28)(22,23)(24,26)(25,29)(27,32)(30,31)(35,41)(39,42)(43,47)(46,48);(2,5)(7,9)(8,12)(15,16)(18,23)(20,29)(21,24)(27,30)(33,34)(35,39)(36,44)(37,38)(40,45)(41,42)(43,48)(46,47);(3,10)(7,9)(11,13)(15,16)(18,23)(19,26)(25,28)(27,30)(34,38)(39,42)(40,44)(46,48);(1,3)(2,7)(4,13)(5,9)(6,10)(8,16)(11,14)(12,15)(17,19,22,26)(18,24,23,21)(20,30,29,27)(25,31,28,32)(33,34)(35,42)(36,40)(37,38)(39,41)(43,48)(44,45)(46,47);(1,4,6,14)(2,8,5,12)(3,11,10,13)(7,15,9,16)(17,20,22,29)(18,25,23,28)(19,27,26,30)(21,31,24,32)(33,35)(34,39)(36,43)(37,41)(38,42)(40,46)(44,48)(45,47);(1,5,6,2)(3,9,10,7)(4,12,14,8)(11,16,13,15)(17,21,22,24)(18,26,23,19)(20,31,29,32)(25,30,28,27)(33,36)(34,40)(35,43)(37,45)(38,44)(39,46)(41,47)(42,48)Polycyclic presentation:
2^9|a^2=gh,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8731974);