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 | |
4039 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,10)(3,8,5,6)(4,9,12,16)(11,15,14,13)(17,18)(19,25)(20,28)(21,24)(22,27)(23,29)(26,30)(31,32)(33,34)(35,43)(36,41)(37,42)(38,39)(40,44)(45,48)(46,47);(1,3,7,5)(2,8,10,6)(4,11,12,14)(9,15,16,13)(17,18,21,24)(19,22,27,25)(20,23,29,28)(26,30,32,31)(35,44)(37,45)(40,43)(42,48);(3,5)(4,12)(6,8)(9,16)(17,18,21,24)(19,25,27,22)(20,28,29,23)(26,30,32,31)(33,35,39,44)(34,40,38,43)(36,37,47,45)(41,48,46,42);(1,4)(2,9)(3,11)(5,14)(6,13)(7,12)(8,15)(10,16)(17,19,21,27)(18,22,24,25)(20,26,29,32)(23,30,28,31)(33,36,39,47)(34,41,38,46)(35,45,44,37)(40,42,43,48);(1,5,7,3)(2,6,10,8)(4,11,12,14)(9,15,16,13)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,29)(30,32)(33,37,39,45)(34,42,38,48)(35,36,44,47)(40,46,43,41);(1,6)(2,3)(4,13)(5,10)(7,8)(9,11)(12,15)(14,16)(17,20)(18,23)(19,26)(21,29)(22,30)(24,28)(25,31)(27,32)(33,38,39,34)(35,40,44,43)(36,46,47,41)(37,48,45,42)Polycyclic presentation:
2^9|a^2=ghi,b^2=hi,c^2=h,d^2=h,e^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8914621);