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 | #52018 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4051 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 56 points:
(1,2,6,8)(3,5,4,7)(26,31)(29,41)(32,35)(33,42)(37,51)(40,52)(44,47)(50,56);(1,3,6,4)(2,7,8,5)(9,10,14,17)(11,12,19,15)(13,16,22,24)(18,20,21,23)(28,39)(33,42)(36,49)(40,52)(44,47)(46,53)(50,56)(54,55);(9,11)(10,15)(12,17)(13,18)(14,19)(16,23)(20,24)(21,22)(25,26,30,31)(27,35,38,32)(28,33,39,42)(29,34,41,43)(36,47,49,44)(37,48,51,45)(40,46,52,53)(50,55,56,54);(2,8)(3,4)(9,12,14,15)(10,11,17,19)(13,20,22,23)(16,18,24,21)(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);(1,4,6,3)(2,7,8,5)(9,11)(10,12)(13,21)(14,19)(15,17)(16,23)(18,22)(20,24)(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);(1,5,6,7)(2,4,8,3)(9,13)(10,16)(11,18)(12,20)(14,22)(15,23)(17,24)(19,21)(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=gi,c^2=h,d^2=hi,e^2=gh,f^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, 10154431);