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 | |
4184 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)(3,5)(4,8)(6,7)(9,13)(10,12)(11,14)(15,16)(17,18)(19,23)(20,25)(21,26)(22,24)(27,30)(28,32)(29,31)(33,34,37,40)(35,41,43,38)(36,39,46,45)(42,48,44,47);(4,11)(8,14)(10,15)(12,16)(17,19,22,26)(18,23,24,21)(20,28,31,27)(25,32,29,30)(33,35,37,43)(34,38,40,41)(36,44,46,42)(39,48,45,47);(2,7)(3,9)(8,14)(10,15)(18,24)(19,26)(25,29)(27,28)(33,35,37,43)(34,38,40,41)(36,42,46,44)(39,47,45,48);(1,3,6,9)(2,5,7,13)(4,10,11,15)(8,12,14,16)(17,18)(19,21)(20,29)(22,24)(23,26)(25,31)(27,30)(28,32)(33,35)(34,38)(36,44)(37,43)(39,48)(40,41)(42,46)(45,47);(1,4,6,11)(2,8,7,14)(3,10,9,15)(5,12,13,16)(17,20,22,31)(18,25,24,29)(19,27,26,28)(21,32,23,30)(33,36)(34,39)(35,42)(37,46)(38,47)(40,45)(41,48)(43,44);(1,5)(2,9)(3,7)(4,12)(6,13)(8,15)(10,14)(11,16)(17,21)(18,19)(20,30)(22,23)(24,26)(25,28)(27,29)(31,32)(33,34)(35,38)(36,45)(37,40)(39,46)(41,43)(42,48)(44,47)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=g,d^2=ghi,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10154273);