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 | |
4131 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,5,10,8)(4,12,14,9)(11,15,16,13)(18,22)(19,25)(24,27)(26,31)(33,34,39,43)(35,40,46,48)(36,38,47,41)(37,45,44,42);(2,7)(5,8)(9,12)(13,15)(17,18,21,22)(19,23,25,29)(20,27,28,24)(26,32,31,30)(33,35,39,46)(34,40,43,48)(36,44,47,37)(38,42,41,45);(2,7)(3,10)(9,12)(11,16)(19,25)(23,29)(26,31)(30,32)(33,36,39,47)(34,41,43,38)(35,37,46,44)(40,42,48,45);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,13)(14,16)(17,19,21,25)(18,23,22,29)(20,26,28,31)(24,30,27,32)(33,36)(34,38)(35,44)(37,46)(39,47)(40,42)(41,43)(45,48);(1,4,6,14)(2,9,7,12)(3,11,10,16)(5,13,8,15)(17,18)(19,23)(20,24)(21,22)(25,29)(26,30)(27,28)(31,32)(33,37,39,44)(34,42,43,45)(35,36,46,47)(38,40,41,48);(1,5,6,8)(2,3,7,10)(4,13,14,15)(9,11,12,16)(17,20,21,28)(18,24,22,27)(19,26,25,31)(23,30,29,32)(33,38)(34,36)(35,45)(37,40)(39,41)(42,46)(43,47)(44,48)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=i,d^2=ghi,e^2=h,f^2=g,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, 8662495);