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 | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4134 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 56 points:
(1,2,5,6)(3,7,8,4)(9,10)(11,16)(12,21)(13,14)(15,18)(17,24)(19,23)(20,22)(26,31)(27,35)(29,43)(33,39)(37,40)(41,54)(46,50)(55,56);(2,6)(4,7)(9,11,15,12)(10,16,18,21)(13,22,17,19)(14,20,24,23)(26,31)(32,36)(33,39)(34,44)(45,47)(46,50)(48,52)(55,56);(1,2)(3,4)(5,6)(7,8)(9,12,15,11)(10,16,18,21)(13,22,17,19)(14,23,24,20)(25,26)(27,36)(28,39)(29,44)(30,31)(32,35)(33,42)(34,43)(37,45)(38,46)(40,47)(41,48)(49,50)(51,56)(52,54)(53,55);(1,3)(2,4)(5,8)(6,7)(25,27)(26,32)(28,40)(29,38)(30,35)(31,36)(33,47)(34,46)(37,42)(39,45)(41,53)(43,49)(44,50)(48,56)(51,54)(52,55);(1,4)(2,3)(5,7)(6,8)(9,13,15,17)(10,14,18,24)(11,19,12,22)(16,23,21,20)(25,28,30,42)(26,33,31,39)(27,37,35,40)(29,41,43,54)(32,45,36,47)(34,48,44,52)(38,51,49,53)(46,55,50,56);(1,3)(2,4)(5,8)(6,7)(9,14)(10,17)(11,20)(12,23)(13,18)(15,24)(16,19)(21,22)(25,29,30,43)(26,34,31,44)(27,38,35,49)(28,41,42,54)(32,46,36,50)(33,48,39,52)(37,51,40,53)(45,55,47,56)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=hi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8965107);