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 | |
4334 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)(3,6)(4,8)(5,9)(7,10)(11,14)(12,13)(15,16)(17,18,22,21)(19,25,27,23)(20,24,28,30)(26,32,29,31)(33,34)(35,40)(36,44)(37,38)(39,43)(41,45)(42,46)(47,48);(1,3)(2,6)(4,11)(5,10)(7,9)(8,14)(12,15)(13,16)(17,19)(18,23)(20,26)(21,25)(22,27)(24,31)(28,29)(30,32)(34,38)(39,40)(41,44)(46,47);(3,10)(6,7)(11,16)(14,15)(20,28)(24,30)(26,29)(31,32)(33,35,37,43)(34,39,38,40)(36,42,45,48)(41,46,44,47);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,20)(18,24)(19,26)(21,30)(22,28)(23,31)(25,32)(27,29)(33,35)(34,40)(36,42)(37,43)(38,39)(41,47)(44,46)(45,48);(1,2)(3,6)(4,12)(5,9)(7,10)(8,13)(11,15)(14,16)(17,19)(18,25)(20,29)(21,23)(22,27)(24,31)(26,28)(30,32)(33,34)(35,39)(36,44)(37,38)(40,43)(41,45)(42,47)(46,48);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,12)(10,16)(17,21,22,18)(19,25,27,23)(20,30,28,24)(26,32,29,31)(33,36,37,45)(34,41,38,44)(35,42,43,48)(39,46,40,47)Polycyclic presentation:
2^9|a^2=gh,c^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9073982);