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 | |
4068 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)(3,5)(4,8)(6,7)(9,13)(10,12)(11,14)(15,16)(17,18)(19,22)(20,25)(21,26)(23,24)(27,32)(28,30)(29,31)(33,34,40,38)(35,43,39,41)(36,42,37,46)(44,48,45,47);(4,11)(8,14)(10,15)(12,16)(17,19,23,27)(18,22,24,32)(20,28,29,21)(25,30,31,26)(33,35,40,39)(34,41,38,43)(36,44,37,45)(42,47,46,48);(2,7)(3,9)(8,14)(10,15)(18,24)(19,27)(20,29)(26,30)(33,36,40,37)(34,42,38,46)(35,44,39,45)(41,47,43,48);(1,3,6,9)(2,5,7,13)(4,10,11,15)(8,12,14,16)(17,20)(18,25)(19,28)(21,27)(22,30)(23,29)(24,31)(26,32)(33,37,40,36)(34,42,38,46)(35,45,39,44)(41,47,43,48);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21,23,28)(18,26,24,30)(19,29,27,20)(22,31,32,25)(33,38,40,34)(35,43,39,41)(36,46,37,42)(44,48,45,47);(1,5)(2,9)(3,7)(4,12)(6,13)(8,15)(10,14)(11,16)(17,22)(18,27)(19,24)(20,30)(21,25)(23,32)(26,29)(28,31)(33,39,40,35)(34,41,38,43)(36,45,37,44)(42,47,46,48)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=g,d^2=ghi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10078881);