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 |
| prime radical | #50072 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3901 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)(3,7)(4,8)(5,6)(9,11)(10,15)(12,14)(13,16)(17,18,22,21)(19,23,28,26)(20,24,29,30)(25,31,27,32)(33,34,39,44)(35,42,37,40)(36,38,43,41)(45,48,47,46);(1,3,6,9)(2,7,5,11)(4,10,12,13)(8,15,14,16)(17,19)(18,23)(20,27)(21,26)(22,28)(24,32)(25,29)(30,31)(33,35,39,37)(34,40,44,42)(36,47,43,45)(38,48,41,46);(2,5)(3,9)(4,12)(15,16)(17,20,22,29)(18,24,21,30)(19,25,28,27)(23,31,26,32)(33,36,39,43)(34,41,44,38)(35,45,37,47)(40,48,42,46);(1,4)(2,8)(3,10)(5,14)(6,12)(7,15)(9,13)(11,16)(17,21,22,18)(19,26,28,23)(20,24,29,30)(25,31,27,32)(33,37,39,35)(34,42,44,40)(36,47,43,45)(38,48,41,46);(1,3)(2,7)(4,13)(5,11)(6,9)(8,16)(10,12)(14,15)(17,20)(18,24)(19,27)(21,30)(22,29)(23,32)(25,28)(26,31)(33,38,39,41)(34,43,44,36)(35,46,37,48)(40,47,42,45);(1,5,6,2)(3,11,9,7)(4,14,12,8)(10,16,13,15)(17,19)(18,23)(20,27)(21,26)(22,28)(24,32)(25,29)(30,31)(33,34)(35,40)(36,41)(37,42)(38,43)(39,44)(45,48)(46,47)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=ghi,d^2=ghi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8901213);