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 | C22×D4 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3973 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,11)(3,8,13,6)(4,9,14,15)(5,10,12,16)(17,18,23,21)(19,27,29,24)(20,30,28,25)(22,26,32,31)(34,39)(35,40)(37,45)(43,47);(1,3)(2,8)(4,5)(6,11)(7,13)(9,10)(12,14)(15,16)(17,19,23,29)(18,24,21,27)(20,22,28,32)(25,26,30,31)(33,34)(35,36)(37,41)(38,39)(40,44)(42,43)(45,46)(47,48);(1,4)(2,9)(3,12)(5,13)(6,10)(7,14)(8,16)(11,15)(17,18,23,21)(19,24,29,27)(20,25,28,30)(22,26,32,31)(33,35,38,40)(34,36,39,44)(37,42,45,48)(41,43,46,47);(1,5,7,12)(2,10,11,16)(3,4,13,14)(6,15,8,9)(17,20)(18,25)(19,22)(21,30)(23,28)(24,26)(27,31)(29,32)(33,34)(35,36)(37,46)(38,39)(40,44)(41,45)(42,47)(43,48);(1,4)(2,9)(3,5)(6,16)(7,14)(8,10)(11,15)(12,13)(17,21,23,18)(19,27,29,24)(20,25,28,30)(22,26,32,31)(33,36)(34,40)(35,39)(37,47)(38,44)(41,42)(43,45)(46,48);(1,6,7,8)(2,3,11,13)(4,10,14,16)(5,9,12,15)(17,22)(18,26)(19,28)(20,29)(21,31)(23,32)(24,30)(25,27)(33,37)(34,41)(35,42)(36,43)(38,45)(39,46)(40,48)(44,47)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=g,d^2=i,e^2=gh,f^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9140034);