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 | #48673 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4103 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)(11,19)(12,17)(13,18)(14,15)(16,20)(21,24)(22,23)(26,27)(28,38)(30,44)(33,36)(39,41)(42,45)(47,51)(55,56);(1,2)(3,7)(4,6)(5,8)(10,15)(11,20)(17,22)(18,21)(25,26,31,27)(28,35,38,32)(29,36,43,33)(30,34,44,37)(39,46,41,48)(40,51,52,47)(42,50,45,49)(53,55,54,56);(1,3)(2,5)(4,8)(6,7)(25,27,31,26)(28,32,38,35)(29,36,43,33)(30,37,44,34)(39,46,41,48)(40,47,52,51)(42,50,45,49)(53,55,54,56);(2,6)(5,7)(9,11)(10,16)(12,18)(13,22)(14,20)(15,19)(17,24)(21,23)(25,28,31,38)(26,32,27,35)(29,41,43,39)(30,40,44,52)(33,48,36,46)(34,47,37,51)(42,54,45,53)(49,56,50,55);(1,2)(3,5)(4,6)(7,8)(9,12,14,23)(10,17,15,22)(11,21,20,18)(13,19,24,16)(25,29,31,43)(26,33,27,36)(28,39,38,41)(30,45,44,42)(32,46,35,48)(34,50,37,49)(40,54,52,53)(47,56,51,55);(1,3)(2,7)(4,8)(5,6)(9,13)(10,18)(11,22)(12,16)(14,24)(15,21)(17,20)(19,23)(25,30)(26,34)(27,37)(28,40)(29,42)(31,44)(32,47)(33,49)(35,51)(36,50)(38,52)(39,53)(41,54)(43,45)(46,55)(48,56)Polycyclic presentation:
2^9|b^2=h,c^2=h,d^2=h,e^2=g,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, 10125804);