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 | #47723 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4057 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)(3,5)(4,7)(6,8)(10,15)(13,21)(16,22)(20,24)(25,26,31,28)(27,35,36,32)(29,39,42,33)(30,40,43,34)(37,44,49,47)(38,45,50,48)(41,46,53,52)(51,55,56,54);(1,2,6,8)(3,5,7,4)(12,19)(16,22)(18,23)(20,24)(27,36)(30,43)(32,35)(34,40)(37,49)(41,53)(44,47)(46,52);(1,3,6,7)(2,4,8,5)(9,10,14,15)(11,13,17,21)(12,16,19,22)(18,20,23,24)(25,27)(26,32)(28,35)(29,37)(30,38)(31,36)(33,44)(34,45)(39,47)(40,48)(41,51)(42,49)(43,50)(46,54)(52,55)(53,56);(1,4,6,5)(2,3,8,7)(9,11,14,17)(10,13,15,21)(12,18,19,23)(16,20,22,24)(25,28,31,26)(27,32,36,35)(29,39,42,33)(30,40,43,34)(37,44,49,47)(38,45,50,48)(41,52,53,46)(51,54,56,55);(1,2)(3,4)(5,7)(6,8)(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,29,31,42)(26,33,28,39)(27,37,36,49)(30,41,43,53)(32,44,35,47)(34,46,40,52)(38,51,50,56)(45,54,48,55);(1,5,6,4)(2,7,8,3)(9,13)(10,17)(11,15)(12,20)(14,21)(16,23)(18,22)(19,24)(25,30)(26,34)(27,38)(28,40)(29,41)(31,43)(32,45)(33,46)(35,48)(36,50)(37,51)(39,52)(42,53)(44,54)(47,55)(49,56)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=ghi,d^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8664514);