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 | |
3741 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,10)(3,8,13,4)(5,6,9,15)(11,12,16,14)(18,23)(20,29)(24,30)(27,32)(33,34,39,38)(35,42,37,40)(36,41,46,45)(43,48,44,47);(1,3,7,13)(2,8,10,4)(5,11,9,16)(6,12,15,14)(19,26)(24,30)(27,32)(28,31)(33,35,39,37)(34,40,38,42)(36,44,46,43)(41,48,45,47);(1,4,7,8)(2,3,10,13)(5,14,9,12)(6,11,15,16)(17,18,22,23)(19,24,26,30)(20,25,29,21)(27,31,32,28)(33,36,39,46)(34,41,38,45)(35,43,37,44)(40,47,42,48);(1,5)(2,6)(3,11)(4,14)(7,9)(8,12)(10,15)(13,16)(17,19,22,26)(18,24,23,30)(20,27,29,32)(21,28,25,31)(33,37,39,35)(34,42,38,40)(36,44,46,43)(41,48,45,47);(1,4)(2,3)(5,14)(6,11)(7,8)(9,12)(10,13)(15,16)(17,20)(18,25)(19,27)(21,23)(22,29)(24,31)(26,32)(28,30)(33,34,39,38)(35,40,37,42)(36,41,46,45)(43,47,44,48);(1,6)(2,9)(3,12)(4,11)(5,10)(7,15)(8,16)(13,14)(17,21,22,25)(18,20,23,29)(19,28,26,31)(24,27,30,32)(33,38,39,34)(35,42,37,40)(36,45,46,41)(43,48,44,47)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=hi,d^2=gh,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8440655);