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 | |
4097 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,7)(3,9)(8,14)(10,15)(33,34,37,38)(35,41,43,39)(36,40,45,44)(42,47,48,46);(1,2,6,7)(3,5,9,13)(4,8,11,14)(10,12,15,16)(17,18,22,23)(19,24,28,31)(20,25,29,21)(26,27,32,30)(34,38)(39,41)(40,44)(46,47);(4,11)(8,14)(10,15)(12,16)(18,23)(20,29)(24,31)(26,32)(33,35,37,43)(34,39,38,41)(36,42,45,48)(40,46,44,47);(1,3)(2,5)(4,10)(6,9)(7,13)(8,12)(11,15)(14,16)(17,19)(18,24)(20,26)(21,30)(22,28)(23,31)(25,27)(29,32)(33,34)(35,39)(36,44)(37,38)(40,45)(41,43)(42,47)(46,48);(1,4,6,11)(2,8,7,14)(3,10,9,15)(5,12,13,16)(17,20)(18,25)(19,26)(21,23)(22,29)(24,27)(28,32)(30,31)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,47)(45,48);(1,5)(2,9)(3,7)(4,12)(6,13)(8,15)(10,14)(11,16)(17,21,22,25)(18,20,23,29)(19,27,28,30)(24,32,31,26)(33,36,37,45)(34,40,38,44)(35,42,43,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=h,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8456743);