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 | |
4237 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,9,10,4)(5,7,13,14)(11,15,16,12)(17,18,22,23)(19,25,27,21)(20,28,29,24)(26,31,32,30)(33,34,39,45)(35,40,47,48)(36,42,46,41)(37,43,44,38);(1,2)(3,4)(5,7)(6,8)(9,10)(11,12)(13,14)(15,16)(17,19,22,27)(18,21,23,25)(20,26,29,32)(24,31,28,30)(33,35,39,47)(34,40,45,48)(36,44,46,37)(38,41,43,42);(3,10)(4,9)(5,13)(7,14)(18,23)(20,29)(21,25)(26,32)(33,36,39,46)(34,41,45,42)(35,37,47,44)(38,48,43,40);(1,3,6,10)(2,4,8,9)(5,11,13,16)(7,12,14,15)(17,20)(18,24)(19,26)(21,31)(22,29)(23,28)(25,30)(27,32)(33,36)(34,42)(35,44)(37,47)(38,40)(39,46)(41,45)(43,48);(1,4)(2,3)(5,12)(6,9)(7,11)(8,10)(13,15)(14,16)(17,21)(18,19)(20,30)(22,25)(23,27)(24,32)(26,28)(29,31)(33,37,39,44)(34,43,45,38)(35,46,47,36)(40,41,48,42);(1,5,6,13)(2,7,8,14)(3,11,10,16)(4,12,9,15)(17,18)(19,21)(20,24)(22,23)(25,27)(26,31)(28,29)(30,32)(33,38,39,43)(34,44,45,37)(35,41,47,42)(36,40,46,48)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=i,d^2=ghi,e^2=i,f^2=gh,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, 8922505);