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 | |
3743 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,9,11,4)(5,13,14,8)(10,12,16,15)(18,23)(21,29)(24,30)(28,32)(34,39)(40,46)(41,47)(42,48);(2,7)(4,9)(8,13)(12,15)(20,27)(24,30)(26,31)(28,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,48)(44,46)(45,47);(1,2)(3,4)(5,8)(6,7)(9,11)(10,12)(13,14)(15,16)(17,18,22,23)(19,21,25,29)(20,24,27,30)(26,28,31,32);(1,3)(2,4)(5,10)(6,11)(7,9)(8,12)(13,15)(14,16)(17,19,22,25)(18,21,23,29)(20,26,27,31)(24,28,30,32)(33,35,38,44)(34,40,39,46)(36,37,45,43)(41,42,47,48);(1,4)(2,3)(5,12)(6,9)(7,11)(8,10)(13,16)(14,15)(17,20,22,27)(18,24,23,30)(19,26,25,31)(21,28,29,32)(33,36,38,45)(34,41,39,47)(35,43,44,37)(40,48,46,42);(1,5,6,14)(2,8,7,13)(3,10,11,16)(4,12,9,15)(17,21)(18,25)(19,23)(20,28)(22,29)(24,31)(26,30)(27,32)(33,37,38,43)(34,42,39,48)(35,36,44,45)(40,41,46,47)Polycyclic presentation:
2^9|a^2=i,c^2=hi,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9143618);