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 | |
4049 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,4)(5,13)(6,7)(8,9)(10,14)(11,15)(12,16)(17,18)(19,20)(21,29)(22,23)(24,28)(25,30)(26,32)(27,31)(34,39)(37,45)(40,41)(43,44);(2,7)(3,9)(10,13)(11,16)(17,19,22,24)(18,20,23,28)(21,26,30,31)(25,27,29,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,48)(45,46);(2,7)(3,9)(10,13)(11,16)(18,23)(19,24)(25,29)(26,31)(33,35,38,36)(34,40,39,41)(37,43,45,44)(42,47,46,48);(1,3)(2,8)(4,7)(5,11)(6,9)(10,15)(12,13)(14,16)(17,19)(18,20)(21,26)(22,24)(23,28)(25,27)(29,32)(30,31)(33,34)(35,41)(36,40)(37,42)(38,39)(43,48)(44,47)(45,46);(1,4,6,8)(2,9,7,3)(5,12,14,15)(10,16,13,11)(17,20)(18,24)(19,23)(21,27)(22,28)(25,31)(26,29)(30,32)(33,36,38,35)(34,41,39,40)(37,44,45,43)(42,48,46,47);(1,5)(2,10)(3,11)(4,12)(6,14)(7,13)(8,15)(9,16)(17,21,22,30)(18,25,23,29)(19,26,24,31)(20,27,28,32)(33,37,38,45)(34,42,39,46)(35,43,36,44)(40,47,41,48)Polycyclic presentation:
2^9|b^2=i,c^2=ghi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8066101);