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 | |
4212 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)(3,8,14,9)(4,5,11,10)(12,13,15,16)(18,22)(20,28)(23,25)(26,32)(33,34,38,41)(35,39,44,47)(36,37,45,40)(42,43,48,46);(2,7)(5,10)(8,9)(13,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,35,38,44)(34,39,41,47)(36,42,45,48)(37,43,40,46);(1,3,6,14)(2,8,7,9)(4,12,11,15)(5,13,10,16)(17,18,21,22)(19,23,27,25)(20,24,28,29)(26,30,32,31)(33,36,38,45)(34,40,41,37)(35,42,44,48)(39,46,47,43);(1,3)(2,9)(4,15)(5,13)(6,14)(7,8)(10,16)(11,12)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,36)(34,37)(35,42)(38,45)(39,43)(40,41)(44,48)(46,47);(1,4,6,11)(2,10,7,5)(3,12,14,15)(8,16,9,13)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,35)(34,39)(36,42)(37,43)(38,44)(40,46)(41,47)(45,48);(1,5,6,10)(2,11,7,4)(3,13,14,16)(8,15,9,12)(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)(33,37)(34,36)(35,43)(38,40)(39,42)(41,45)(44,46)(47,48)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=ghi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dghi,e^d=egi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9297768);