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 | |
4205 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(4,12)(8,13)(10,16)(14,15)(17,18)(19,27)(20,30)(21,28)(22,24)(23,29)(25,31)(26,32)(34,36)(37,45)(39,42)(43,47);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,19,22,29)(18,23,24,27)(20,26,31,28)(21,25,32,30)(34,36)(35,41)(40,44)(43,47);(2,6)(7,9)(8,13)(14,15)(18,24)(20,31)(23,27)(26,28)(33,34,38,36)(35,42,41,39)(37,40,45,44)(43,48,47,46);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,20,22,31)(18,25,24,30)(19,28,29,26)(21,27,32,23)(33,35,38,41)(34,39,36,42)(37,43,45,47)(40,46,44,48);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,19)(18,23)(20,28)(21,25)(22,29)(24,27)(26,31)(30,32)(33,36,38,34)(35,42,41,39)(37,44,45,40)(43,48,47,46);(1,4)(2,8)(3,10)(5,12)(6,13)(7,14)(9,15)(11,16)(17,21)(18,26)(19,25)(20,27)(22,32)(23,31)(24,28)(29,30)(33,37,38,45)(34,40,36,44)(35,43,41,47)(39,46,42,48)Polycyclic presentation:
2^9|b^2=g,c^2=gh,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8019612);