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 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3992 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(5,13)(8,14)(10,15)(11,16)(17,18,22,26)(19,27,30,23)(20,24,31,29)(21,28,32,25)(33,34,38,39)(35,41,36,40)(37,45,46,42)(43,47,44,48);(1,2)(3,4)(5,8)(6,7)(9,12)(10,11)(13,14)(15,16)(17,19,22,30)(18,23,26,27)(20,28,31,25)(21,29,32,24)(34,39)(40,41)(42,45)(47,48);(2,7)(3,9)(8,14)(10,15)(17,18,22,26)(19,23,30,27)(20,24,31,29)(21,25,32,28)(33,35,38,36)(34,40,39,41)(37,43,46,44)(42,47,45,48);(1,3,6,9)(2,4,7,12)(5,10,13,15)(8,11,14,16)(17,18)(19,23)(20,29)(21,25)(22,26)(24,31)(27,30)(28,32)(33,34)(35,41)(36,40)(37,42)(38,39)(43,48)(44,47)(45,46);(1,4)(2,3)(5,11)(6,12)(7,9)(8,10)(13,16)(14,15)(17,20)(18,24)(19,28)(21,23)(22,31)(25,30)(26,29)(27,32)(33,36,38,35)(34,41,39,40)(37,44,46,43)(42,48,45,47);(1,5)(2,8)(3,10)(4,11)(6,13)(7,14)(9,15)(12,16)(17,21)(18,25)(19,29)(20,23)(22,32)(24,30)(26,28)(27,31)(33,37)(34,42)(35,43)(36,44)(38,46)(39,45)(40,47)(41,48)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=g,d^2=i,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8066792);