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 | #47066 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4075 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 56 points:
(9,10)(11,14)(12,15)(13,16)(17,20)(18,22)(19,23)(21,24)(26,31)(27,35)(29,42)(33,39)(37,47)(40,53)(45,50)(54,56);(1,2)(3,7)(4,6)(5,8)(10,11)(15,17)(16,18)(23,24)(25,26,30,31)(27,32,35,36)(28,39,41,33)(29,34,42,43)(37,49,47,44)(38,45,48,50)(40,52,53,46)(51,56,55,54);(1,3)(2,5)(4,8)(6,7)(25,26)(27,36)(28,33)(29,34)(30,31)(32,35)(37,49)(38,50)(39,41)(40,46)(42,43)(44,47)(45,48)(51,56)(52,53)(54,55);(2,6)(5,7)(9,11,14,10)(12,15,20,17)(13,18,22,16)(19,23,21,24)(25,27)(26,32)(28,37)(29,38)(30,35)(31,36)(33,44)(34,45)(39,49)(40,51)(41,47)(42,48)(43,50)(46,54)(52,56)(53,55);(1,2)(3,5)(4,6)(7,8)(9,12,14,20)(10,15,11,17)(13,21,22,19)(16,24,18,23)(25,28,30,41)(26,33,31,39)(27,37,35,47)(29,40,42,53)(32,44,36,49)(34,46,43,52)(38,51,48,55)(45,54,50,56);(1,3)(2,7)(4,8)(5,6)(9,13,14,22)(10,16,11,18)(12,19,20,21)(15,23,17,24)(25,29)(26,34)(27,38)(28,40)(30,42)(31,43)(32,45)(33,46)(35,48)(36,50)(37,51)(39,52)(41,53)(44,54)(47,55)(49,56)Polycyclic presentation:
2^9|b^2=h,d^2=gh,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10127591);