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 | |
4054 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,7)(3,9)(10,14)(11,16)(17,18)(19,24)(20,26)(21,22)(23,28)(25,29)(27,32)(30,31)(33,34,38,37)(35,39,44,42)(36,40,45,46)(41,47,43,48);(1,2)(3,8)(4,9)(5,10)(6,7)(11,15)(12,16)(13,14)(17,19,23,30)(18,24,28,31)(20,27,29,21)(22,26,32,25)(33,35)(34,39)(36,43)(37,42)(38,44)(40,48)(41,45)(46,47);(3,9)(4,8)(5,13)(10,14)(17,20,23,29)(18,25,28,26)(19,21,30,27)(22,24,32,31)(33,36,38,45)(34,40,37,46)(35,41,44,43)(39,47,42,48);(1,3)(2,8)(4,7)(5,11)(6,9)(10,15)(12,14)(13,16)(17,20)(18,26)(19,27)(21,30)(22,31)(23,29)(24,32)(25,28)(33,37,38,34)(35,42,44,39)(36,40,45,46)(41,47,43,48);(1,4,6,8)(2,9,7,3)(5,12,13,15)(10,16,14,11)(17,21,23,27)(18,22,28,32)(19,29,30,20)(24,25,31,26)(33,36)(34,40)(35,43)(37,46)(38,45)(39,48)(41,44)(42,47);(1,5,6,13)(2,10,7,14)(3,11,9,16)(4,12,8,15)(17,22)(18,27)(19,26)(20,31)(21,28)(23,32)(24,29)(25,30)(33,35)(34,39)(36,43)(37,42)(38,44)(40,48)(41,45)(46,47)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=i,d^2=hi,e^2=gh,f^2=g,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, 8900176);