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 | |
4002 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,9,10,7)(4,12,13,8)(11,14,16,15)(21,29)(25,30)(26,31)(27,32)(33,34,39,35)(36,43,46,40)(37,41,38,42)(44,48,45,47);(2,6)(3,10)(8,12)(11,16)(17,18)(19,20)(21,25)(22,23)(24,28)(26,27)(29,30)(31,32)(34,35)(40,43)(41,42)(47,48);(2,6)(7,9)(8,12)(14,15)(18,23)(19,24)(25,30)(26,31)(33,35,39,34)(36,43,46,40)(37,42,38,41)(44,48,45,47);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,19,22,24)(18,20,23,28)(21,26,29,31)(25,27,30,32)(33,36,39,46)(34,40,35,43)(37,44,38,45)(41,47,42,48);(1,3,5,10)(2,7,6,9)(4,11,13,16)(8,14,12,15)(17,20)(18,24)(19,23)(21,27)(22,28)(25,31)(26,30)(29,32)(33,37,39,38)(34,41,35,42)(36,44,46,45)(40,47,43,48);(1,4,5,13)(2,8,6,12)(3,11,10,16)(7,14,9,15)(17,21)(18,25)(19,26)(20,27)(22,29)(23,30)(24,31)(28,32)(33,38,39,37)(34,42,35,41)(36,45,46,44)(40,48,43,47)Polycyclic presentation:
2^9|a^2=g,c^2=gh,d^2=i,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8019766);