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 | |
4241 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)(3,10)(4,5)(6,7)(8,13)(9,15)(11,12)(14,16)(17,18,22,25)(19,26,20,23)(21,24,29,31)(27,32,28,30)(33,34)(35,37)(36,43)(38,46)(39,41)(40,47)(42,44)(45,48);(2,7)(4,9)(8,10)(11,14)(17,19,22,20)(18,23,25,26)(21,27,29,28)(24,30,31,32)(33,35,39,43)(34,37,41,36)(38,42,47,48)(40,45,46,44);(2,7)(5,15)(8,10)(12,16)(21,29)(24,31)(27,28)(30,32)(33,36,39,37)(34,35,41,43)(38,45,47,44)(40,48,46,42);(1,3,6,13)(2,8,7,10)(4,14,9,11)(5,12,15,16)(17,18)(19,23)(20,26)(21,24)(22,25)(27,30)(28,32)(29,31)(33,37,39,36)(34,35,41,43)(38,45,47,44)(40,42,46,48);(1,4,6,9)(2,5,7,15)(3,11,13,14)(8,16,10,12)(17,20,22,19)(18,23,25,26)(21,28,29,27)(24,30,31,32)(33,35)(34,37)(36,41)(38,42)(39,43)(40,45)(44,46)(47,48);(1,5)(2,9)(3,12)(4,7)(6,15)(8,11)(10,14)(13,16)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)(33,38)(34,40)(35,42)(36,44)(37,45)(39,47)(41,46)(43,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=i,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8805756);