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 | |
4310 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,8,11,14)(4,9,13,16)(5,12,15,10)(17,18,22,20)(19,23,26,24)(21,25,29,28)(27,30,32,31)(33,34,39,42)(35,36,37,40)(38,46,47,41)(43,48,45,44);(2,7)(4,13)(8,14)(10,12)(17,19)(18,23)(20,24)(21,27)(22,26)(25,30)(28,31)(29,32)(33,35)(34,36)(37,39)(38,43)(40,42)(41,44)(45,47)(46,48);(2,7)(3,11)(9,16)(10,12)(19,26)(23,24)(27,32)(30,31)(33,36,39,40)(34,35,42,37)(38,44,47,48)(41,43,46,45);(1,3)(2,8)(4,10)(5,9)(6,11)(7,14)(12,13)(15,16)(17,19)(18,24)(20,23)(21,27)(22,26)(25,31)(28,30)(29,32)(33,36)(34,35)(37,42)(38,44)(39,40)(41,43)(45,46)(47,48);(1,4,6,13)(2,9,7,16)(3,12,11,10)(5,8,15,14)(17,20,22,18)(19,24,26,23)(21,25,29,28)(27,30,32,31)(33,37,39,35)(34,40,42,36)(38,43,47,45)(41,44,46,48);(1,5,6,15)(2,10,7,12)(3,9,11,16)(4,14,13,8)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)(33,38)(34,41)(35,43)(36,44)(37,45)(39,47)(40,48)(42,46)Polycyclic presentation:
2^9|a^2=gi,c^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9110960);