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 | |
4125 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,9,10,7)(4,8,12,13)(11,15,16,14)(17,18,22,23)(19,26,25,21)(20,28,30,24)(27,29,31,32)(35,41)(39,42)(43,47)(46,48);(2,6)(4,12)(7,9)(11,16)(18,23)(21,26)(24,28)(29,32)(33,34)(35,42)(36,44)(37,38)(39,41)(40,45)(43,46)(47,48);(3,10)(7,9)(11,16)(14,15)(18,23)(19,25)(24,28)(27,31)(34,38)(39,42)(40,44)(46,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,16)(13,15)(17,19,22,25)(18,21,23,26)(20,27,30,31)(24,29,28,32)(33,34)(35,39)(36,40)(37,38)(41,42)(43,46)(44,45)(47,48);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,20,22,30)(18,24,23,28)(19,27,25,31)(21,29,26,32)(33,35,37,41)(34,39,38,42)(36,43,45,47)(40,46,44,48);(1,4,5,12)(2,8,6,13)(3,11,10,16)(7,14,9,15)(17,21)(18,25)(19,23)(20,29)(22,26)(24,31)(27,28)(30,32)(33,36)(34,40)(35,43)(37,45)(38,44)(39,46)(41,47)(42,48)Polycyclic presentation:
2^9|a^2=g,d^2=h,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8663388);