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 | |
4099 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)(3,4)(5,6)(7,8)(9,10)(11,21)(12,17)(13,23)(14,18)(15,19)(16,20)(22,24)(26,31)(29,41)(32,35)(33,42)(37,51)(40,52)(44,47)(50,56);(2,6)(3,7)(9,11,16,19)(10,15,20,21)(12,14,13,24)(17,22,23,18)(28,39)(33,42)(36,49)(40,52)(44,47)(46,53)(50,56)(54,55);(1,3,5,7)(2,4,6,8)(9,12,16,13)(10,17,20,23)(11,14,19,24)(15,22,21,18)(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);(1,4)(2,7)(3,6)(5,8)(9,13,16,12)(10,17,20,23)(11,14,19,24)(15,18,21,22)(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);(9,14)(10,18)(11,13)(12,19)(15,17)(16,24)(20,22)(21,23)(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)(10,19)(11,20)(12,22)(13,18)(14,23)(16,21)(17,24)(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|b^2=gi,c^2=gh,d^2=ghi,e^2=h,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, 10154414);