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 | |
4037 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)(3,10)(4,8)(5,7)(6,9)(11,16)(12,14)(13,15)(17,18)(19,21)(20,23)(22,28)(24,26)(25,30)(27,31)(29,32)(33,34)(35,38)(36,43)(37,40)(39,44)(41,45)(42,48)(46,47);(1,3,6,5)(2,7,9,10)(4,11,13,12)(8,14,15,16)(17,19)(18,24)(20,21)(22,27)(23,26)(25,31)(28,29)(30,32)(35,41)(38,45)(42,47)(46,48);(18,20)(19,26)(25,28)(27,32)(33,34,37,40)(35,38,41,45)(36,39,44,43)(42,46,47,48);(1,4,6,13)(2,8,9,15)(3,11,5,12)(7,14,10,16)(17,20,23,18)(19,21,26,24)(22,28,30,25)(27,29,32,31)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48);(1,5,6,3)(2,7,9,10)(4,12,13,11)(8,14,15,16)(17,21)(18,19)(20,26)(22,29)(23,24)(25,27)(28,32)(30,31)(33,35,37,41)(34,38,40,45)(36,42,44,47)(39,46,43,48);(1,2)(3,7)(4,8)(5,10)(6,9)(11,14)(12,16)(13,15)(17,22,23,30)(18,25,20,28)(19,27,26,32)(21,29,24,31)(33,36,37,44)(34,39,40,43)(35,42,41,47)(38,46,45,48)Polycyclic presentation:
2^9|b^2=gi,c^2=hi,d^2=g,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,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, 9637854);