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 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3837 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,9)(3,7,10,5)(4,8,12,16)(11,15,13,14)(17,18,23,20)(19,24,27,28)(21,25,31,29)(22,30,32,26)(33,34,40,38)(35,43,47,41)(36,37,42,39)(44,45,48,46);(1,3)(2,7)(4,11)(5,9)(6,10)(8,15)(12,13)(14,16)(17,19)(18,24)(20,28)(21,30)(22,29)(23,27)(25,32)(26,31)(33,35,40,47)(34,41,38,43)(36,48,42,44)(37,45,39,46);(3,10)(4,12)(5,7)(8,16)(18,20)(19,27)(21,31)(22,32)(33,36)(34,39)(35,44)(37,38)(40,42)(41,45)(43,46)(47,48);(1,3)(2,7)(4,13)(5,9)(6,10)(8,14)(11,12)(15,16)(17,20,23,18)(19,28,27,24)(21,25,31,29)(22,30,32,26)(33,37,40,39)(34,36,38,42)(35,45,47,46)(41,48,43,44);(1,4)(2,8)(3,11)(5,14)(6,12)(7,15)(9,16)(10,13)(17,21)(18,25)(19,26)(20,29)(22,24)(23,31)(27,30)(28,32)(33,38,40,34)(35,41,47,43)(36,37,42,39)(44,45,48,46);(1,5,6,7)(2,3,9,10)(4,14,12,15)(8,11,16,13)(17,22,23,32)(18,26,20,30)(19,29,27,25)(21,24,31,28)(33,39,40,37)(34,42,38,36)(35,46,47,45)(41,44,43,48)Polycyclic presentation:
2^9|a^2=i,b^2=g,d^2=h,e^2=g,f^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8927410);