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 | |
4055 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)(3,4)(5,13)(6,7)(8,12)(9,14)(10,16)(11,15)(17,18,23,25)(19,20,21,24)(22,30,31,26)(27,32,29,28)(33,34,39,40)(35,41,36,38)(37,45,47,42)(43,46,44,48);(1,3,6,8)(2,4,7,12)(5,10,14,15)(9,11,13,16)(17,19,23,21)(18,24,25,20)(22,27,31,29)(26,32,30,28)(34,40)(35,36)(42,45)(43,44);(2,7)(3,8)(9,13)(10,15)(18,25)(20,24)(26,30)(28,32)(33,35,39,36)(34,38,40,41)(37,43,47,44)(42,46,45,48);(1,3)(2,4)(5,10)(6,8)(7,12)(9,11)(13,16)(14,15)(17,20,23,24)(18,19,25,21)(22,28,31,32)(26,27,30,29)(33,36,39,35)(34,41,40,38)(37,44,47,43)(42,48,45,46);(1,4)(2,8)(3,7)(5,11)(6,12)(9,15)(10,13)(14,16)(17,21,23,19)(18,20,25,24)(22,29,31,27)(26,28,30,32)(33,37,39,47)(34,42,40,45)(35,43,36,44)(38,46,41,48);(1,5,6,14)(2,9,7,13)(3,10,8,15)(4,11,12,16)(17,22,23,31)(18,26,25,30)(19,27,21,29)(20,28,24,32)(33,38,39,41)(34,35,40,36)(37,46,47,48)(42,43,45,44)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=ghi,d^2=g,e^2=g,f^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, 8066080);