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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3939 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)(3,9)(4,13)(5,14)(6,8)(7,12)(10,11)(15,16)(18,23)(19,25)(26,31)(27,32)(33,34,38,37)(35,39,41,43)(36,40,44,45)(42,46,47,48);(1,3,6,12)(2,7,8,9)(4,10,14,16)(5,11,13,15)(17,18)(19,24)(20,25)(21,26)(22,23)(27,29)(28,32)(30,31)(35,41)(39,43)(42,47)(46,48);(2,8)(4,14)(7,9)(10,16)(33,35,38,41)(34,39,37,43)(36,42,44,47)(40,46,45,48);(1,4)(2,5)(3,10)(6,14)(7,11)(8,13)(9,15)(12,16)(17,19,22,25)(18,24,23,20)(21,27,30,32)(26,29,31,28);(1,3)(2,9)(4,10)(5,15)(6,12)(7,8)(11,13)(14,16)(17,20,22,24)(18,25,23,19)(21,29,30,28)(26,27,31,32)(33,36)(34,40)(35,42)(37,45)(38,44)(39,46)(41,47)(43,48);(1,5)(2,4)(3,11)(6,13)(7,10)(8,14)(9,16)(12,15)(17,21,22,30)(18,26,23,31)(19,27,25,32)(20,28,24,29)(33,37,38,34)(35,43,41,39)(36,40,44,45)(42,46,47,48)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=i,d^2=gh,e^2=gh,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9619172);