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:
(1,2,6,10)(3,7,11,15)(4,13,14,8)(5,12,16,9)(17,18)(19,26)(20,24)(21,29)(22,23)(25,31)(27,28)(30,32)(33,34,39,43)(35,42,37,40)(36,41,47,38)(44,46,45,48);(1,3)(2,7)(4,9)(5,13)(6,11)(8,16)(10,15)(12,14)(17,19,23,24)(18,20,22,26)(21,27,31,32)(25,28,29,30)(33,35)(34,40)(36,44)(37,39)(38,46)(41,48)(42,43)(45,47);(3,11)(4,14)(7,15)(8,13)(18,22)(20,26)(25,29)(28,30)(33,36,39,47)(34,41,43,38)(35,44,37,45)(40,48,42,46);(1,4)(2,8)(3,12)(5,7)(6,14)(9,11)(10,13)(15,16)(17,20,23,26)(18,24,22,19)(21,30,31,28)(25,27,29,32)(33,36)(34,41)(35,45)(37,44)(38,43)(39,47)(40,46)(42,48);(1,5)(2,9)(3,8)(4,15)(6,16)(7,14)(10,12)(11,13)(17,21)(18,25)(19,27)(20,28)(22,29)(23,31)(24,32)(26,30)(33,37,39,35)(34,42,43,40)(36,45,47,44)(38,48,41,46);(1,3)(2,7)(4,12)(5,8)(6,11)(9,14)(10,15)(13,16)(17,22,23,18)(19,26,24,20)(21,29,31,25)(27,30,32,28)(33,38,39,41)(34,36,43,47)(35,46,37,48)(40,44,42,45)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=gh,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8841520);