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 | |
4122 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)(4,5)(6,8)(9,10)(11,15)(12,16)(13,14)(17,22)(18,21)(19,20)(23,24)(26,27)(30,43)(32,35)(33,36)(39,53)(41,44)(45,49)(51,54);(2,5)(6,7)(10,14)(15,21)(16,19)(22,23)(25,26,31,27)(28,32,40,35)(29,33,42,36)(30,34,43,37)(38,45,52,49)(39,46,53,50)(41,47,44,48)(51,55,54,56);(1,2)(3,6)(4,5)(7,8)(25,27,31,26)(28,32,40,35)(29,36,42,33)(30,37,43,34)(38,45,52,49)(39,46,53,50)(41,48,44,47)(51,55,54,56);(1,3,4,8)(2,6,5,7)(9,11,13,18)(10,15,14,21)(12,17,20,24)(16,22,19,23)(25,28,31,40)(26,32,27,35)(29,38,42,52)(30,39,43,53)(33,45,36,49)(34,46,37,50)(41,51,44,54)(47,55,48,56);(9,10)(11,15)(12,19)(13,14)(16,20)(17,23)(18,21)(22,24)(25,29,31,42)(26,33,27,36)(28,38,40,52)(30,44,43,41)(32,45,35,49)(34,48,37,47)(39,54,53,51)(46,56,50,55);(9,12)(10,16)(11,17)(13,20)(14,19)(15,22)(18,24)(21,23)(25,30)(26,34)(27,37)(28,39)(29,41)(31,43)(32,46)(33,47)(35,50)(36,48)(38,51)(40,53)(42,44)(45,55)(49,56)(52,54)Polycyclic presentation:
2^9|b^2=h,c^2=h,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9818186);