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:
(1,2,6,11)(3,9,14,7)(4,5,10,8)(12,16,15,13)(17,18,22,21)(19,23,25,27)(20,24,28,29)(26,30,31,32)(33,34)(35,41)(36,46)(37,39)(38,42)(40,43)(44,48)(45,47);(1,3)(2,7)(4,12)(5,13)(6,14)(8,16)(9,11)(10,15)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,31)(29,32)(33,35,40,37)(34,41,43,39)(36,45,38,44)(42,48,46,47);(4,10)(5,8)(12,15)(13,16)(19,25)(20,28)(23,27)(24,29)(33,36,40,38)(34,42,43,46)(35,44,37,45)(39,48,41,47);(1,4,6,10)(2,8,11,5)(3,12,14,15)(7,16,9,13)(17,20,22,28)(18,24,21,29)(19,26,25,31)(23,30,27,32)(33,37,40,35)(34,39,43,41)(36,44,38,45)(42,47,46,48);(1,3)(2,9)(4,15)(5,13)(6,14)(7,11)(8,16)(10,12)(17,21,22,18)(19,27,25,23)(20,24,28,29)(26,30,31,32)(33,38,40,36)(34,42,43,46)(35,45,37,44)(39,48,41,47);(1,5,6,8)(2,10,11,4)(3,13,14,16)(7,15,9,12)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,31)(29,32)(33,39)(34,35)(36,47)(37,43)(38,48)(40,41)(42,45)(44,46)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=h,d^2=gi,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8829748);