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 | |
4314 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,9)(3,7,10,4)(5,8,14,16)(11,15,13,12)(17,18)(19,24)(20,30)(21,26)(22,23)(25,29)(27,32)(28,31)(33,34)(35,36)(37,45)(38,42)(39,40)(41,46)(43,47)(44,48);(1,3,6,10)(2,7,9,4)(5,13,14,11)(8,12,16,15)(18,23)(21,30)(24,25)(28,32)(33,34,38,42)(35,39,40,36)(37,45,46,41)(43,44,48,47);(3,10)(4,7)(11,13)(12,15)(17,19,22,29)(18,24,23,25)(20,27,26,31)(21,28,30,32)(33,35,38,40)(34,39,42,36)(37,43,46,48)(41,47,45,44);(1,4)(2,3)(5,12)(6,7)(8,11)(9,10)(13,16)(14,15)(17,19)(18,25)(20,31)(21,28)(22,29)(23,24)(26,27)(30,32)(33,35)(34,39)(36,42)(37,43)(38,40)(41,47)(44,45)(46,48);(1,2)(3,7)(4,10)(5,8)(6,9)(11,15)(12,13)(14,16)(17,20,22,26)(18,21,23,30)(19,27,29,31)(24,28,25,32)(33,36,38,39)(34,40,42,35)(37,44,46,47)(41,48,45,43);(1,5)(2,8)(3,11)(4,12)(6,14)(7,15)(9,16)(10,13)(17,21)(18,26)(19,28)(20,23)(22,30)(24,31)(25,27)(29,32)(33,37,38,46)(34,41,42,45)(35,43,40,48)(36,44,39,47)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=g,e^2=g,f^2=h,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, 8735837);