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 | |
3987 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,5)(3,8,14,13)(4,10,15,9)(6,16,12,11)(17,18,22,25)(19,26,20,23)(21,29,30,24)(27,31,28,32)(33,34,39,42)(35,36,40,37)(38,41,46,48)(43,44,47,45);(1,3,7,14)(2,8,5,13)(4,12,15,6)(9,16,10,11)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,31)(29,30)(33,35,39,40)(34,37,42,36)(38,43,46,47)(41,45,48,44);(1,4)(2,9)(3,12)(5,10)(6,14)(7,15)(8,16)(11,13)(17,19,22,20)(18,23,25,26)(21,27,30,28)(24,31,29,32)(33,36,39,37)(34,40,42,35)(38,44,46,45)(41,47,48,43);(1,4)(2,10)(3,12)(5,9)(6,14)(7,15)(8,11)(13,16)(17,18)(19,23)(20,26)(21,24)(22,25)(27,31)(28,32)(29,30)(33,35)(34,36)(37,42)(38,43)(39,40)(41,44)(45,48)(46,47);(1,5,7,2)(3,13,14,8)(4,10,15,9)(6,16,12,11)(17,20,22,19)(18,23,25,26)(21,28,30,27)(24,31,29,32)(33,37,39,36)(34,35,42,40)(38,44,46,45)(41,47,48,43);(1,6,7,12)(2,11,5,16)(3,4,14,15)(8,9,13,10)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,29)(26,32)(33,38,39,46)(34,41,42,48)(35,43,40,47)(36,44,37,45)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=g,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10413479);