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 | |
4122 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,7,9,4)(5,13,15,10)(11,12,14,16)(17,18,21,23)(19,25,27,22)(20,28,30,24)(26,31,29,32)(33,34,38,42)(35,36,40,39)(37,45,47,41)(43,46,48,44);(1,3)(2,7)(4,8)(5,14)(6,9)(10,12)(11,15)(13,16)(17,19,21,27)(18,22,23,25)(20,29,30,26)(24,32,28,31)(33,35,38,40)(34,39,42,36)(37,43,47,48)(41,46,45,44);(2,8)(3,9)(10,13)(11,14)(18,23)(22,25)(24,28)(31,32)(33,36,38,39)(34,40,42,35)(37,44,47,46)(41,48,45,43);(1,3)(2,7)(4,8)(5,11)(6,9)(10,16)(12,13)(14,15)(17,18)(19,22)(20,28)(21,23)(24,30)(25,27)(26,32)(29,31)(33,36)(34,35)(37,46)(38,39)(40,42)(41,48)(43,45)(44,47);(1,4,6,7)(2,9,8,3)(5,12,15,16)(10,14,13,11)(17,20,21,30)(18,24,23,28)(19,26,27,29)(22,31,25,32)(33,37,38,47)(34,41,42,45)(35,43,40,48)(36,44,39,46);(1,5,6,15)(2,10,8,13)(3,11,9,14)(4,12,7,16)(17,19)(18,22)(20,26)(21,27)(23,25)(24,31)(28,32)(29,30)(33,35)(34,36)(37,43)(38,40)(39,42)(41,44)(45,46)(47,48)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=ghi,e^2=gi,f^2=i,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=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8796556);