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 | |
4050 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,3)(4,7,11,9)(5,10,13,8)(12,15,16,14)(17,18)(19,28)(20,31)(21,22)(23,24)(25,29)(26,30)(27,32)(35,38)(40,42)(43,45)(47,48);(1,2,6,3)(4,9,11,7)(5,10,13,8)(12,14,16,15)(18,24)(19,29)(21,27)(26,31)(33,34,39,36)(35,40,38,42)(37,44,46,41)(43,48,45,47);(2,3)(4,11)(8,10)(12,16)(17,19)(18,25)(20,27)(21,30)(22,31)(23,29)(24,28)(26,32)(33,35,39,38)(34,40,36,42)(37,43,46,45)(41,47,44,48);(1,3,6,2)(4,9,11,7)(5,10,13,8)(12,15,16,14)(17,20,23,30)(18,26,24,31)(19,27,29,21)(22,25,32,28)(33,36,39,34)(35,42,38,40)(37,44,46,41)(43,48,45,47);(1,4,6,11)(2,7,3,9)(5,12,13,16)(8,14,10,15)(17,21,23,27)(18,22,24,32)(19,30,29,20)(25,31,28,26)(33,37,39,46)(34,41,36,44)(35,43,38,45)(40,47,42,48);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)(17,22)(18,27)(19,31)(20,25)(21,24)(23,32)(26,29)(28,30)(33,38,39,35)(34,42,36,40)(37,45,46,43)(41,48,44,47)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=hi,d^2=h,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8680160);