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 | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4026 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 56 points:
(1,2)(3,7)(4,6)(5,8)(9,10,15,16)(11,19,20,17)(12,23,24,13)(14,21,22,18)(25,26,31,35)(27,32,36,29)(28,39,43,33)(30,34,47,50)(37,44,40,41)(38,48,46,45)(42,52,55,49)(51,56,53,54);(1,2,4,6)(3,5,8,7)(10,16)(11,20)(13,23)(18,21)(27,36)(29,32)(37,40)(38,46)(41,44)(45,48)(51,53)(54,56);(1,2)(3,5)(4,6)(7,8)(9,11,15,20)(10,17,16,19)(12,21,24,18)(13,22,23,14)(25,27)(26,32)(28,40)(29,35)(30,46)(31,36)(33,44)(34,45)(37,43)(38,47)(39,41)(42,51)(48,50)(49,54)(52,56)(53,55);(1,3)(2,5)(4,8)(6,7)(9,12)(10,13)(11,21)(14,19)(15,24)(16,23)(17,22)(18,20)(25,28,31,43)(26,33,35,39)(27,37,36,40)(29,44,32,41)(30,42,47,55)(34,49,50,52)(38,51,46,53)(45,56,48,54);(9,13)(10,12)(11,22)(14,20)(15,23)(16,24)(17,21)(18,19)(25,29,31,32)(26,27,35,36)(28,41,43,44)(30,45,47,48)(33,40,39,37)(34,38,50,46)(42,54,55,56)(49,53,52,51);(9,14,15,22)(10,18,16,21)(11,13,20,23)(12,19,24,17)(25,30)(26,34)(27,38)(28,42)(29,45)(31,47)(32,48)(33,49)(35,50)(36,46)(37,51)(39,52)(40,53)(41,54)(43,55)(44,56)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=h,d^2=gi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8839719);