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 | #58547 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4053 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 56 points:
(1,2,5,3)(4,7,8,6)(10,14)(15,20)(16,21)(22,24)(26,31)(28,39)(29,41)(32,36)(37,47)(38,48)(46,53)(54,56);(2,3)(6,7)(11,17)(15,20)(18,23)(22,24)(26,31)(27,35)(28,39)(34,43)(38,48)(40,52)(44,49)(54,56);(1,3,5,2)(4,7,8,6)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,26)(27,36)(28,33)(29,34)(30,31)(32,35)(37,49)(38,50)(39,42)(40,46)(41,43)(44,47)(45,48)(51,56)(52,53)(54,55);(1,4)(2,6)(3,7)(5,8)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,27,30,35)(26,32,31,36)(28,37,39,47)(29,38,41,48)(33,44,42,49)(34,45,43,50)(40,51,52,55)(46,54,53,56);(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,28,30,39)(26,33,31,42)(27,37,35,47)(29,40,41,52)(32,44,36,49)(34,46,43,53)(38,51,48,55)(45,54,50,56);(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,29,30,41)(26,34,31,43)(27,38,35,48)(28,40,39,52)(32,45,36,50)(33,46,42,53)(37,51,47,55)(44,54,49,56)Polycyclic presentation:
2^9|a^2=g,c^2=g,d^2=gi,e^2=gi,f^2=h,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, 8021601);