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 | |
4020 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)(3,5,9,13)(4,8,11,14)(10,12,15,16)(17,18)(19,22)(20,27)(21,23)(24,29)(25,31)(26,32)(28,30)(33,34)(35,36)(37,45)(38,39)(40,43)(41,46)(42,47)(44,48);(4,11)(8,14)(10,15)(12,16)(17,19,21,25)(18,22,23,31)(20,28,29,26)(24,32,27,30)(34,39)(36,40)(37,46)(42,48);(2,7)(3,9)(8,14)(10,15)(18,23)(19,25)(24,27)(26,28)(33,35,38,43)(34,40,39,36)(37,42,46,48)(41,47,45,44);(1,3)(2,5)(4,10)(6,9)(7,13)(8,12)(11,15)(14,16)(17,18)(19,22)(20,24)(21,23)(25,31)(26,30)(27,29)(28,32)(33,35)(34,40)(36,39)(37,42)(38,43)(41,47)(44,45)(46,48);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,19)(18,22)(20,26)(21,25)(23,31)(24,30)(27,32)(28,29)(33,36,38,40)(34,35,39,43)(37,44,46,47)(41,42,45,48);(1,5)(2,3)(4,12)(6,13)(7,9)(8,10)(11,16)(14,15)(17,20)(18,24)(19,26)(21,29)(22,30)(23,27)(25,28)(31,32)(33,37)(34,41)(35,42)(36,44)(38,46)(39,45)(40,47)(43,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=i,c^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9150351);