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 | |
4145 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,11,12,8)(4,9,14,13)(5,15,16,10)(17,18)(19,23)(20,24)(21,22)(25,27)(26,31)(28,30)(29,32)(34,39)(35,40)(41,46)(42,47);(2,7)(4,14)(8,11)(10,15)(17,19,22,25)(18,23,21,27)(20,26,28,29)(24,31,30,32)(33,34,38,39)(35,36,40,43)(37,41,45,46)(42,44,47,48);(2,7)(3,12)(4,14)(5,16)(18,21)(19,25)(20,28)(31,32)(35,40)(36,43)(37,45)(41,46);(1,3,6,12)(2,8,7,11)(4,15,14,10)(5,13,16,9)(17,20)(18,24)(19,26)(21,30)(22,28)(23,31)(25,29)(27,32)(33,35)(34,36)(37,42)(38,40)(39,43)(41,44)(45,47)(46,48);(1,4)(2,9)(3,10)(5,11)(6,14)(7,13)(8,16)(12,15)(17,19)(18,23)(20,29)(21,27)(22,25)(24,32)(26,28)(30,31)(33,36)(34,40)(35,39)(37,44)(38,43)(41,47)(42,46)(45,48);(1,5,6,16)(2,10,7,15)(3,13,12,9)(4,11,14,8)(17,21,22,18)(19,27,25,23)(20,30,28,24)(26,32,29,31)(33,37)(34,41)(35,42)(36,44)(38,45)(39,46)(40,47)(43,48)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,d^2=hi,f^2=h,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, 8894178);