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×#540 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4081 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 56 points:
(1,2,4,6)(3,5,7,8)(10,14)(11,17)(12,20)(21,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,45)(36,38)(37,46)(39,41)(40,47)(42,48)(43,44)(49,55)(50,51)(52,54)(53,56);(1,3,4,7)(2,5,6,8)(9,10,15,14)(11,18,17,13)(12,16,20,22)(19,24,23,21)(26,31)(32,36)(33,39)(34,44)(45,51)(46,48)(47,52)(55,56);(1,2)(3,5)(4,6)(7,8)(9,11)(10,13)(12,19)(14,18)(15,17)(16,21)(20,23)(22,24)(25,27)(26,32)(28,35)(29,42)(30,38)(31,36)(33,45)(34,48)(37,43)(39,51)(40,53)(41,50)(44,46)(47,56)(49,54)(52,55);(9,12)(10,16)(11,19)(13,21)(14,22)(15,20)(17,23)(18,24)(25,28,30,41)(26,33,31,39)(27,35,38,50)(29,40,43,54)(32,45,36,51)(34,47,44,52)(37,49,42,53)(46,55,48,56);(1,3)(2,5)(4,7)(6,8)(9,13,15,18)(10,17,14,11)(12,21,20,24)(16,23,22,19)(25,26)(27,36)(28,39)(29,34)(30,31)(32,38)(33,41)(35,45)(37,48)(40,52)(42,46)(43,44)(47,54)(49,55)(50,51)(53,56);(1,2)(3,5)(4,6)(7,8)(9,14,15,10)(11,18,17,13)(12,22,20,16)(19,24,23,21)(25,29,30,43)(26,34,31,44)(27,37,38,42)(28,40,41,54)(32,46,36,48)(33,47,39,52)(35,49,50,53)(45,55,51,56)Polycyclic presentation:
2^9|a^2=hi,b^2=h,d^2=gi,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8947293);