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 | |
4073 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,5)(3,9,11,7)(4,12,14,8)(10,15,13,16)(18,22)(19,25)(24,29)(26,31)(33,34,39,38)(35,41,36,40)(37,42,46,45)(43,48,44,47);(1,3,6,11)(2,7,5,9)(4,13,14,10)(8,16,12,15)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,35,39,36)(34,40,38,41)(37,44,46,43)(42,48,45,47);(2,5)(7,9)(8,12)(15,16)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,35,39,36)(34,41,38,40)(37,43,46,44)(42,48,45,47);(1,3)(2,7)(4,10)(5,9)(6,11)(8,15)(12,16)(13,14)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,36,39,35)(34,41,38,40)(37,44,46,43)(42,48,45,47);(1,4,6,14)(2,8,5,12)(3,10,11,13)(7,15,9,16)(17,19,21,25)(18,23,22,28)(20,26,27,31)(24,30,29,32)(33,37,39,46)(34,42,38,45)(35,43,36,44)(40,47,41,48);(1,5,6,2)(3,9,11,7)(4,12,14,8)(10,16,13,15)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,38,39,34)(35,41,36,40)(37,45,46,42)(43,48,44,47)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=gh,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8109552);