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 | |
4155 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,12,13,8)(4,14,11,6)(5,10,15,16)(17,18,22,26)(19,23,27,31)(20,24,30,28)(21,25,32,29)(33,34,38,36)(35,40,43,39)(37,44,45,41)(42,46,48,47);(1,3)(2,8)(4,5)(6,10)(7,13)(9,12)(11,15)(14,16)(17,19)(18,23)(20,29)(21,24)(22,27)(25,30)(26,31)(28,32)(33,35,38,43)(34,39,36,40)(37,42,45,48)(41,46,44,47);(2,9)(3,13)(4,11)(10,16)(19,27)(20,30)(23,31)(24,28)(33,35,38,43)(34,40,36,39)(37,42,45,48)(41,47,44,46);(1,4,7,11)(2,6,9,14)(3,5,13,15)(8,10,12,16)(17,20)(18,24)(19,25)(21,31)(22,30)(23,32)(26,28)(27,29)(33,35)(34,39)(36,40)(37,42)(38,43)(41,46)(44,47)(45,48);(1,5)(2,10)(3,11)(4,13)(6,12)(7,15)(8,14)(9,16)(17,21)(18,25)(19,28)(20,23)(22,32)(24,27)(26,29)(30,31)(33,36,38,34)(35,39,43,40)(37,41,45,44)(42,47,48,46);(1,6)(2,11)(3,10)(4,9)(5,12)(7,14)(8,15)(13,16)(17,18)(19,23)(20,24)(21,29)(22,26)(25,32)(27,31)(28,30)(33,37,38,45)(34,41,36,44)(35,42,43,48)(39,46,40,47)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=gh,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10219579);