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 | #2670 | 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) = 128 |
A minimal faithful permutation representation — 6 generators on 56 points:
(9,10)(11,22)(12,18)(13,23)(14,19)(15,20)(16,21)(17,24)(25,26)(27,35)(28,39)(29,43)(30,31)(32,36)(33,42)(34,44)(37,45)(38,46)(40,47)(41,48)(49,50)(51,56)(52,54)(53,55);(1,2)(3,7)(4,6)(5,8)(9,11,16,24)(10,17,21,22)(12,14,13,20)(15,23,19,18)(26,31)(27,36)(33,39)(34,43)(37,40)(38,50)(48,52)(51,53);(1,3)(2,5)(4,8)(6,7)(9,12,16,13)(10,18,21,23)(11,14,24,20)(15,22,19,17)(25,27)(26,32)(28,40)(29,38)(30,36)(31,35)(33,47)(34,46)(37,42)(39,45)(41,53)(43,49)(44,50)(48,56)(51,54)(52,55);(2,6)(5,7)(9,13,16,12)(10,18,21,23)(11,14,24,20)(15,17,19,22)(25,28)(26,33)(27,37)(29,41)(30,42)(31,39)(32,45)(34,48)(35,47)(36,40)(38,51)(43,52)(44,54)(46,55)(49,56)(50,53);(1,2)(3,5)(4,6)(7,8)(9,14)(10,19)(11,13)(12,24)(15,21)(16,20)(17,18)(22,23)(25,26)(27,32)(28,33)(29,34)(30,31)(35,36)(37,45)(38,46)(39,42)(40,47)(41,48)(43,44)(49,50)(51,55)(52,54)(53,56);(1,3)(2,7)(4,8)(5,6)(9,15,16,19)(10,20,21,14)(11,23,24,18)(12,22,13,17)(25,29,30,44)(26,34,31,43)(27,38,36,50)(28,41,42,54)(32,46,35,49)(33,48,39,52)(37,51,40,53)(45,55,47,56)Polycyclic presentation:
2^9|b^2=gh,c^2=gh,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10125834);