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 | |
4101 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,11,12,8)(4,13,14,9)(5,10,15,16)(17,18)(19,26)(20,28)(21,23)(22,25)(24,29)(27,31)(30,32)(33,34)(35,37)(36,45)(38,40)(39,42)(41,44)(43,47)(46,48);(2,7)(3,12)(9,13)(10,16)(17,19,22,21)(18,23,25,26)(20,27,29,30)(24,31,28,32)(33,35)(34,37)(36,46)(38,43)(39,44)(40,47)(41,42)(45,48);(2,7)(3,12)(9,13)(10,16)(20,29)(24,28)(27,30)(31,32)(33,36,39,38)(34,40,42,45)(35,43,44,46)(37,48,41,47);(1,3,6,12)(2,8,7,11)(4,10,14,16)(5,9,15,13)(17,20,22,29)(18,24,25,28)(19,27,21,30)(23,31,26,32)(33,37,39,41)(34,35,42,44)(36,47,38,48)(40,46,45,43);(1,4,6,14)(2,9,7,13)(3,10,12,16)(5,8,15,11)(17,21,22,19)(18,23,25,26)(20,30,29,27)(24,31,28,32)(33,35)(34,41)(36,43)(37,42)(38,46)(39,44)(40,47)(45,48);(1,5,6,15)(2,10,7,16)(3,13,12,9)(4,11,14,8)(17,18)(19,23)(20,28)(21,26)(22,25)(24,29)(27,32)(30,31)(33,38,39,36)(34,40,42,45)(35,43,44,46)(37,47,41,48)Polycyclic presentation:
2^9|a^2=hi,b^2=ghi,c^2=i,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8627910);