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 | |
4022 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)(3,6)(4,11)(5,8)(7,12)(9,13)(10,16)(14,15)(17,18,22,25)(19,21,28,31)(20,24,30,23)(26,32,27,29)(33,34,39,37)(35,42,45,40)(36,38,47,41)(43,44,46,48);(3,9)(6,13)(10,15)(14,16)(17,19,22,28)(18,21,25,31)(20,27,30,26)(23,32,24,29)(34,37)(38,41)(40,42)(44,48);(1,2,5,8)(3,6,9,13)(4,7,12,11)(10,14,15,16)(17,20,22,30)(18,23,25,24)(19,26,28,27)(21,29,31,32)(33,35,39,45)(34,40,37,42)(36,46,47,43)(38,44,41,48);(1,2)(3,6)(4,7)(5,8)(9,13)(10,14)(11,12)(15,16)(17,20)(18,24)(19,27)(21,29)(22,30)(23,25)(26,28)(31,32)(33,36,39,47)(34,41,37,38)(35,43,45,46)(40,44,42,48);(1,3)(2,6)(4,10)(5,9)(7,14)(8,13)(11,16)(12,15)(17,19)(18,21)(20,27)(22,28)(23,32)(24,29)(25,31)(26,30)(33,37,39,34)(35,40,45,42)(36,38,47,41)(43,44,46,48);(1,4)(2,7)(3,10)(5,12)(6,14)(8,11)(9,15)(13,16)(17,21)(18,19)(20,29)(22,31)(23,26)(24,27)(25,28)(30,32)(33,38)(34,36)(35,44)(37,47)(39,41)(40,46)(42,43)(45,48)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=h,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9150419);