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 | |
4093 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,12)(3,10,14,7)(4,9,15,8)(5,13,16,11)(17,18,22,25)(19,20,27,23)(21,29,30,24)(26,31,32,28)(33,34)(35,40)(36,47)(37,41)(38,45)(39,43)(42,44)(46,48);(1,3)(2,7)(4,11)(5,8)(6,14)(9,16)(10,12)(13,15)(17,18)(19,20)(21,24)(22,25)(23,27)(26,28)(29,30)(31,32)(33,35,39,46)(34,40,43,48)(36,44,41,38)(37,45,47,42);(1,4)(2,8)(3,11)(5,7)(6,15)(9,12)(10,16)(13,14)(17,19)(18,23)(20,25)(21,26)(22,27)(24,31)(28,29)(30,32)(33,36,39,41)(34,37,43,47)(35,44,46,38)(40,45,48,42);(1,4)(2,9)(3,13)(5,7)(6,15)(8,12)(10,16)(11,14)(17,20,22,23)(18,19,25,27)(21,28,30,31)(24,26,29,32)(33,34)(35,40)(36,37)(38,42)(39,43)(41,47)(44,45)(46,48);(1,3)(2,10)(4,11)(5,9)(6,14)(7,12)(8,16)(13,15)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,31)(30,32)(33,37)(34,41)(35,45)(36,43)(38,40)(39,47)(42,46)(44,48);(1,5)(2,11)(3,8)(4,7)(6,16)(9,14)(10,15)(12,13)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,29)(27,32)(33,38)(34,42)(35,36)(37,40)(39,44)(41,46)(43,45)(47,48)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=gh,d^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10388176);