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 | #55618 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4049 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 56 points:
(1,2,7,5)(3,4,8,6)(10,14)(15,20)(16,21)(22,24)(27,35)(31,36)(37,40)(38,49)(45,47)(46,50)(51,53)(55,56);(2,5)(3,8)(12,19)(16,21)(17,23)(22,24)(25,26,30,34)(27,31,35,36)(28,39,42,32)(29,33,44,43)(37,47,40,45)(38,46,49,50)(41,52,54,48)(51,56,53,55);(1,3)(2,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,22)(18,20)(19,21)(23,24)(25,26)(27,36)(28,39)(29,43)(30,34)(31,35)(32,42)(33,44)(37,45)(38,46)(40,47)(41,48)(49,50)(51,56)(52,54)(53,55);(1,4,7,6)(2,3,5,8)(9,11,13,18)(10,15,14,20)(12,17,19,23)(16,22,21,24)(25,27)(26,31)(28,40)(29,38)(30,35)(32,47)(33,46)(34,36)(37,42)(39,45)(41,53)(43,50)(44,49)(48,56)(51,54)(52,55);(1,5,7,2)(3,4,8,6)(9,12)(10,16)(11,17)(13,19)(14,21)(15,22)(18,23)(20,24)(25,28,30,42)(26,32,34,39)(27,37,35,40)(29,41,44,54)(31,45,36,47)(33,48,43,52)(38,51,49,53)(46,55,50,56);(1,6,7,4)(2,8,5,3)(9,10)(11,15)(12,16)(13,14)(17,22)(18,20)(19,21)(23,24)(25,29)(26,33)(27,38)(28,41)(30,44)(31,46)(32,48)(34,43)(35,49)(36,50)(37,51)(39,52)(40,53)(42,54)(45,55)(47,56)Polycyclic presentation:
2^9|a^2=g,b^2=gi,d^2=h,e^2=i,f^2=g,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, 8961184);