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 | |
4111 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)(17,18,22,21)(19,25,27,23)(20,28,29,24)(26,31,30,32)(33,34,39,45)(35,40,47,48)(36,42,46,41)(37,38,44,43);(2,7)(3,12)(9,13)(10,16)(17,19)(18,23)(20,26)(21,25)(22,27)(24,31)(28,32)(29,30)(33,35,39,47)(34,40,45,48)(36,38,46,43)(37,42,44,41);(2,7)(3,12)(9,13)(10,16)(20,29)(24,28)(26,30)(31,32)(33,36,39,46)(34,41,45,42)(35,43,47,38)(37,40,44,48);(1,3,6,12)(2,8,7,11)(4,10,14,16)(5,9,15,13)(17,20)(18,24)(19,26)(21,28)(22,29)(23,31)(25,32)(27,30)(33,36)(34,42)(35,43)(37,40)(38,47)(39,46)(41,45)(44,48);(1,4,6,14)(2,9,7,13)(3,10,12,16)(5,8,15,11)(17,21,22,18)(19,25,27,23)(20,28,29,24)(26,32,30,31)(33,37)(34,43)(35,42)(36,40)(38,45)(39,44)(41,47)(46,48);(1,5,6,15)(2,10,7,16)(3,13,12,9)(4,11,14,8)(17,19)(18,25)(20,30)(21,23)(22,27)(24,31)(26,29)(28,32)(33,38,39,43)(34,44,45,37)(35,46,47,36)(40,41,48,42)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=i,d^2=hi,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, 8628031);