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 | |
4197 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,10,9,5)(4,12,13,8)(11,14,15,16)(20,28)(24,29)(26,32)(30,31)(34,36)(37,45)(39,42)(43,47);(2,7)(3,9)(4,13)(14,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(34,36)(35,41)(40,44)(43,47);(2,7)(3,9)(8,12)(11,15)(18,22)(23,25)(24,29)(30,31)(33,34,38,36)(35,42,41,39)(37,40,45,44)(43,48,47,46);(1,3,6,9)(2,5,7,10)(4,11,13,15)(8,14,12,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,35,38,41)(34,39,36,42)(37,43,45,47)(40,46,44,48);(1,4)(2,8)(3,11)(5,14)(6,13)(7,12)(9,15)(10,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,36,38,34)(35,42,41,39)(37,44,45,40)(43,48,47,46);(1,5)(2,9)(3,7)(4,14)(6,10)(8,15)(11,12)(13,16)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)(33,37,38,45)(34,40,36,44)(35,43,41,47)(39,46,42,48)Polycyclic presentation:
2^9|a^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, 8019611);