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 | |
4166 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)(3,7)(4,9)(5,13)(6,8)(10,14)(11,12)(15,16)(17,18,22,23)(19,21,26,24)(20,28,30,25)(27,31,32,29)(34,39)(35,36)(42,45)(43,44);(1,3,6,9)(2,7,8,4)(5,11,14,16)(10,15,13,12)(18,23)(19,26)(25,28)(27,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,48)(45,46);(2,8)(3,9)(10,13)(11,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,30)(29,32)(33,34)(35,40)(36,41)(37,45)(38,39)(42,46)(43,48)(44,47);(1,4,6,7)(2,9,8,3)(5,12,14,15)(10,16,13,11)(17,19,22,26)(18,24,23,21)(20,27,30,32)(25,31,28,29)(33,35,38,36)(34,40,39,41)(37,43,46,44)(42,47,45,48);(1,5,6,14)(2,10,8,13)(3,11,9,16)(4,12,7,15)(17,20,22,30)(18,25,23,28)(19,27,26,32)(21,29,24,31)(33,36,38,35)(34,41,39,40)(37,44,46,43)(42,48,45,47);(1,2)(3,7)(4,9)(5,10)(6,8)(11,15)(12,16)(13,14)(17,21,22,24)(18,26,23,19)(20,29,30,31)(25,32,28,27)(33,37,38,46)(34,42,39,45)(35,43,36,44)(40,47,41,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8086860);