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 | |
4264 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)(3,7)(4,11)(5,6)(8,13)(9,16)(10,14)(12,15)(17,18)(19,23)(20,25)(21,26)(22,24)(27,28)(29,31)(30,32)(34,38)(36,44)(39,41)(42,48);(2,6)(7,14)(8,11)(15,16)(17,19,22,26)(18,23,24,21)(20,27,29,32)(25,28,31,30)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48);(2,6)(7,14)(8,11)(15,16)(18,24)(19,26)(25,31)(27,32)(33,34,37,38)(35,39,43,41)(36,40,44,45)(42,46,48,47);(1,2)(3,7)(4,8)(5,6)(9,15)(10,14)(11,13)(12,16)(17,19)(18,21)(20,27)(22,26)(23,24)(25,30)(28,31)(29,32)(33,34)(35,39)(36,40)(37,38)(41,43)(42,46)(44,45)(47,48);(1,3,5,10)(2,7,6,14)(4,12,13,9)(8,16,11,15)(17,20)(18,25)(19,27)(21,30)(22,29)(23,28)(24,31)(26,32)(33,35,37,43)(34,39,38,41)(36,42,44,48)(40,46,45,47);(1,4,5,13)(2,8,6,11)(3,9,10,12)(7,15,14,16)(17,21)(18,19)(20,28)(22,23)(24,26)(25,32)(27,31)(29,30)(33,36,37,44)(34,40,38,45)(35,42,43,48)(39,46,41,47)Polycyclic presentation:
2^9|b^2=gh,c^2=ghi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9846781);