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 | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3890 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)(3,11)(4,13)(5,10)(6,7)(8,12)(9,15)(14,16)(33,34,37,40)(35,38,42,46)(36,43,44,39)(41,47,48,45);(2,7)(3,12)(9,13)(10,14)(17,18,22,23)(19,21,25,30)(20,24,29,31)(26,28,27,32)(33,35,37,42)(34,38,40,46)(36,41,44,48)(39,45,43,47);(2,7)(5,16)(8,11)(9,13)(18,23)(21,30)(24,31)(28,32)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48);(1,3)(2,8)(4,14)(5,13)(6,12)(7,11)(9,16)(10,15)(17,19,22,25)(18,21,23,30)(20,27,29,26)(24,32,31,28)(33,35)(34,38)(36,41)(37,42)(39,45)(40,46)(43,47)(44,48);(1,4,6,15)(2,9,7,13)(3,10,12,14)(5,11,16,8)(17,20)(18,24)(19,26)(21,28)(22,29)(23,31)(25,27)(30,32)(33,36,37,44)(34,39,40,43)(35,41,42,48)(38,45,46,47);(1,5)(2,10)(3,13)(4,11)(6,16)(7,14)(8,15)(9,12)(17,21)(18,25)(19,23)(20,28)(22,30)(24,27)(26,31)(29,32)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,d^2=i,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8202830);