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 | |
4074 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,10)(3,8,5,6)(4,9,12,16)(11,15,14,13)(17,18)(19,27)(20,25)(21,26)(22,23)(24,28)(29,32)(30,31)(33,34,39,44)(35,42,37,40)(36,46,47,41)(38,43,45,48);(1,3,7,5)(2,8,10,6)(4,11,12,14)(9,15,16,13)(19,28)(21,29)(24,27)(26,32)(33,35,39,37)(34,40,44,42)(36,38,47,45)(41,43,46,48);(3,5)(4,12)(6,8)(9,16)(17,19,23,28)(18,24,22,27)(20,21,31,29)(25,32,30,26)(33,36,39,47)(34,41,44,46)(35,45,37,38)(40,48,42,43);(1,4)(2,9)(3,11)(5,14)(6,13)(7,12)(8,15)(10,16)(17,20,23,31)(18,25,22,30)(19,29,28,21)(24,26,27,32)(33,37,39,35)(34,42,44,40)(36,38,47,45)(41,43,46,48);(1,5,7,3)(2,6,10,8)(4,11,12,14)(9,15,16,13)(17,21,23,29)(18,26,22,32)(19,20,28,31)(24,30,27,25)(33,38)(34,43)(35,36)(37,47)(39,45)(40,41)(42,46)(44,48);(1,6)(2,3)(4,13)(5,10)(7,8)(9,11)(12,15)(14,16)(17,22,23,18)(19,24,28,27)(20,30,31,25)(21,32,29,26)(33,34)(35,40)(36,41)(37,42)(38,43)(39,44)(45,48)(46,47)Polycyclic presentation:
2^9|a^2=ghi,b^2=ghi,c^2=h,d^2=h,e^2=gi,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, 8914639);