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 | #58135 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4096 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 56 points:
(3,7)(5,8)(9,10,16,15)(11,19,14,17)(12,22,18,13)(20,21,23,24)(29,41)(34,43)(37,51)(40,52)(45,48)(46,53)(50,56)(54,55);(1,2,4,6)(3,5,7,8)(9,11,16,14)(10,17,15,19)(12,23,18,20)(13,24,22,21)(26,31)(28,39)(32,35)(34,43)(36,49)(40,52)(45,48)(50,56);(1,3)(2,5)(4,7)(6,8)(9,12)(10,13)(11,20)(14,23)(15,22)(16,18)(17,21)(19,24)(25,26)(27,35)(28,33)(29,34)(30,31)(32,38)(36,47)(37,48)(39,42)(40,46)(41,43)(44,49)(45,51)(50,55)(52,53)(54,56);(1,2)(3,5)(4,6)(7,8)(9,13,16,22)(10,18,15,12)(11,21,14,24)(17,23,19,20)(25,27,30,38)(26,32,31,35)(28,36,39,49)(29,37,41,51)(33,44,42,47)(34,45,43,48)(40,50,52,56)(46,54,53,55);(1,3)(2,5)(4,7)(6,8)(9,14,16,11)(10,19,15,17)(12,23,18,20)(13,24,22,21)(25,28,30,39)(26,33,31,42)(27,36,38,49)(29,40,41,52)(32,44,35,47)(34,46,43,53)(37,50,51,56)(45,54,48,55);(9,15,16,10)(11,19,14,17)(12,22,18,13)(20,24,23,21)(25,29)(26,34)(27,37)(28,40)(30,41)(31,43)(32,45)(33,46)(35,48)(36,50)(38,51)(39,52)(42,53)(44,54)(47,55)(49,56)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=i,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8021554);