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 | |
4226 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)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,18)(19,26)(20,30)(21,25)(22,23)(24,27)(28,29)(31,32)(33,34,39,38)(35,42,36,40)(37,45,46,41)(43,47,44,48);(2,6)(7,9)(8,13)(14,15)(18,23)(19,27)(20,25)(29,32)(35,36)(37,46)(40,42)(41,45);(4,12)(8,13)(10,16)(14,15)(18,23)(21,30)(24,26)(29,32)(33,35,39,36)(34,40,38,42)(37,43,46,44)(41,47,45,48);(1,3,5,11)(2,7,6,9)(4,10,12,16)(8,14,13,15)(17,19)(18,24)(20,28)(21,29)(22,27)(23,26)(25,31)(30,32)(33,36,39,35)(34,40,38,42)(37,44,46,43)(41,47,45,48);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,20,22,25)(18,21,23,30)(19,28,27,31)(24,29,26,32)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48);(1,4)(2,8)(3,10)(5,12)(6,13)(7,14)(9,15)(11,16)(17,21)(18,25)(19,29)(20,23)(22,30)(24,31)(26,28)(27,32)(33,38,39,34)(35,40,36,42)(37,45,46,41)(43,47,44,48)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=gh,e^2=hi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8165813);