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 | C2×#553 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3955 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 56 points:
(9,10,14,17)(11,15,19,13)(12,20,22,16)(18,21,24,23)(25,26)(27,31)(28,35)(29,33)(30,37)(32,40)(34,43)(36,41)(38,47)(39,45)(42,49)(44,50)(46,53)(48,52)(51,54)(55,56);(1,2)(3,4)(5,6)(7,8)(9,11,14,19)(10,13,17,15)(12,18,22,24)(16,21,20,23)(26,27)(29,41)(32,35)(34,37)(38,50)(42,53)(45,48)(51,56);(2,6)(3,7)(9,11)(10,15)(12,18)(13,17)(14,19)(16,23)(20,21)(22,24)(25,27,31,26)(28,32,40,35)(29,36,41,33)(30,37,43,34)(38,44,50,47)(39,45,52,48)(42,49,53,46)(51,54,56,55);(3,7)(4,8)(9,12,14,22)(10,16,17,20)(11,18,19,24)(13,21,15,23)(25,28,31,40)(26,32,27,35)(29,38,41,50)(30,39,43,52)(33,44,36,47)(34,45,37,48)(42,51,53,56)(46,54,49,55);(1,3)(2,4)(5,7)(6,8)(25,29,31,41)(26,33,27,36)(28,38,40,50)(30,42,43,53)(32,44,35,47)(34,46,37,49)(39,51,52,56)(45,54,48,55);(1,4)(2,3)(5,8)(6,7)(9,13,14,15)(10,11,17,19)(12,21,22,23)(16,18,20,24)(25,30,31,43)(26,34,27,37)(28,39,40,52)(29,42,41,53)(32,45,35,48)(33,46,36,49)(38,51,50,56)(44,54,47,55)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=h,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,e^d=egi,f^d=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9532649);