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 | |
4114 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)(19,21)(20,28)(22,23)(24,27)(25,31)(26,32)(29,30)(33,34,37,41)(35,38,43,39)(36,44,46,40)(42,48,45,47);(1,3,6,10)(2,7,9,4)(5,13,14,11)(8,12,16,15)(18,23)(21,24)(25,28)(30,32)(33,35)(34,38)(36,45)(37,43)(39,41)(40,48)(42,46)(44,47);(3,10)(4,7)(11,13)(12,15)(17,19,22,27)(18,21,23,24)(20,26,31,29)(25,30,28,32)(33,35,37,43)(34,39,41,38)(36,42,46,45)(40,48,44,47);(1,4)(2,3)(5,12)(6,7)(8,11)(9,10)(13,16)(14,15)(17,19)(18,24)(20,29)(21,23)(22,27)(25,30)(26,31)(28,32)(33,35)(34,38)(36,42)(37,43)(39,41)(40,47)(44,48)(45,46);(1,2)(3,7)(4,10)(5,8)(6,9)(11,15)(12,13)(14,16)(17,20,22,31)(18,25,23,28)(19,26,27,29)(21,30,24,32)(33,34)(35,39)(36,40)(37,41)(38,43)(42,48)(44,46)(45,47);(1,5)(2,8)(3,11)(4,12)(6,14)(7,15)(9,16)(10,13)(17,21,22,24)(18,19,23,27)(20,30,31,32)(25,26,28,29)(33,36)(34,40)(35,42)(37,46)(38,47)(39,48)(41,44)(43,45)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=gh,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, 8735145);