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 | |
4310 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)(3,9)(4,12)(5,6)(7,10)(8,14)(11,15)(13,16)(17,18,21,22)(19,25,27,23)(20,28,29,24)(26,31,30,32)(35,41)(38,42)(43,47)(46,48);(2,6)(7,9)(8,12)(15,16)(18,22)(20,29)(23,25)(26,30)(33,34)(35,38)(36,44)(37,40)(39,45)(41,42)(43,48)(46,47);(3,10)(7,9)(11,13)(15,16)(17,19,21,27)(18,23,22,25)(20,26,29,30)(24,31,28,32)(33,34,37,40)(35,38,41,42)(36,39,45,44)(43,46,47,48);(1,3)(2,7)(4,13)(5,10)(6,9)(8,16)(11,14)(12,15)(17,19)(18,23)(20,30)(21,27)(22,25)(24,32)(26,29)(28,31)(33,34)(35,42)(36,39)(37,40)(38,41)(43,48)(44,45)(46,47);(1,4,5,14)(2,8,6,12)(3,11,10,13)(7,15,9,16)(17,20,21,29)(18,24,22,28)(19,26,27,30)(23,31,25,32)(33,35)(34,38)(36,43)(37,41)(39,46)(40,42)(44,48)(45,47);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,31)(28,29)(30,32)(33,36)(34,39)(35,43)(37,45)(38,46)(40,44)(41,47)(42,48)Polycyclic presentation:
2^9|a^2=h,c^2=hi,e^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, 8736413);