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×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3977 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 56 points:
(2,5)(6,8)(10,16)(11,21)(12,14)(19,24)(25,26,31,29)(27,35,36,32)(28,39,42,33)(30,43,45,34)(37,46,40,48)(38,47,44,50)(41,49,54,52)(51,56,53,55);(3,7)(6,8)(9,10)(11,22)(12,18)(13,24)(14,20)(15,16)(17,21)(19,23)(27,36)(32,35)(37,40)(38,44)(46,48)(47,50)(51,53)(55,56);(1,2)(3,6)(4,5)(7,8)(9,11)(10,17)(12,13)(14,23)(15,21)(16,22)(18,19)(20,24)(25,27)(26,32)(28,40)(29,35)(30,44)(31,36)(33,48)(34,50)(37,42)(38,45)(39,46)(41,51)(43,47)(49,55)(52,56)(53,54);(9,12,15,14)(10,18,16,20)(11,13,21,23)(17,19,22,24)(25,28,31,42)(26,33,29,39)(27,37,36,40)(30,41,45,54)(32,46,35,48)(34,49,43,52)(38,51,44,53)(47,55,50,56);(1,3,4,7)(2,6,5,8)(9,13,15,23)(10,19,16,24)(11,12,21,14)(17,18,22,20)(25,29,31,26)(27,35,36,32)(28,33,42,39)(30,43,45,34)(37,46,40,48)(38,50,44,47)(41,49,54,52)(51,55,53,56);(1,2)(3,6)(4,5)(7,8)(9,14,15,12)(10,20,16,18)(11,23,21,13)(17,24,22,19)(25,30)(26,34)(27,38)(28,41)(29,43)(31,45)(32,47)(33,49)(35,50)(36,44)(37,51)(39,52)(40,53)(42,54)(46,55)(48,56)Polycyclic presentation:
2^9|a^2=gi,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8837702);