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 | |
4028 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,11)(3,8,12,14)(4,9,13,16)(5,6,10,15)(17,18)(19,22)(20,30)(21,32)(23,27)(24,29)(25,31)(26,28)(33,34)(35,41)(36,47)(37,38)(39,43)(40,44)(42,46)(45,48);(1,3,7,12)(2,8,11,14)(4,6,13,15)(5,9,10,16)(17,19)(18,24)(20,21)(22,27)(23,29)(25,32)(26,30)(28,31)(33,35,40,47)(34,41,44,36)(37,43,42,45)(38,39,46,48);(3,12)(4,13)(8,14)(9,16)(17,20,23,31)(18,25,27,30)(19,21,29,28)(22,26,24,32)(33,36,40,41)(34,35,44,47)(37,48,42,39)(38,43,46,45);(1,4)(2,9)(3,6)(5,14)(7,13)(8,10)(11,16)(12,15)(17,20)(18,25)(19,28)(21,29)(22,32)(23,31)(24,26)(27,30)(33,37,40,42)(34,38,44,46)(35,45,47,43)(36,39,41,48);(1,5)(2,6)(3,9)(4,14)(7,10)(8,13)(11,15)(12,16)(17,21,23,28)(18,26,27,32)(19,20,29,31)(22,25,24,30)(33,38)(34,42)(35,39)(36,43)(37,44)(40,46)(41,45)(47,48);(1,6)(2,10)(3,13)(4,12)(5,11)(7,15)(8,16)(9,14)(17,22,23,24)(18,19,27,29)(20,26,31,32)(21,30,28,25)(33,39)(34,43)(35,46)(36,37)(38,47)(40,48)(41,42)(44,45)Polycyclic presentation:
2^9|a^2=ghi,b^2=i,c^2=h,d^2=gh,e^2=g,f^2=g,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, 8619530);