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 | |
3911 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,4,8,11)(5,9,13,14)(10,12,15,16)(17,18,22,23)(19,21,25,29)(20,24,27,30)(26,28,31,32)(33,34,39,41)(35,45,46,40)(36,37,43,42)(38,48,47,44);(2,7)(4,11)(9,14)(12,16)(20,27)(24,30)(26,31)(28,32)(33,35)(34,40)(36,47)(37,48)(38,43)(39,46)(41,45)(42,44);(5,13)(9,14)(10,15)(12,16)(18,23)(19,25)(24,30)(26,31)(34,41)(35,46)(36,43)(44,48);(1,3)(2,4)(5,10)(6,8)(7,11)(9,12)(13,15)(14,16)(17,19)(18,21)(20,26)(22,25)(23,29)(24,28)(27,31)(30,32)(33,36,39,43)(34,42,41,37)(35,47,46,38)(40,44,45,48);(1,4)(2,8)(3,7)(5,12)(6,11)(9,15)(10,14)(13,16)(17,20,22,27)(18,24,23,30)(19,26,25,31)(21,28,29,32)(33,37,39,42)(34,43,41,36)(35,44,46,48)(38,40,47,45);(1,5,6,13)(2,9,7,14)(3,10,8,15)(4,12,11,16)(17,21)(18,19)(20,28)(22,29)(23,25)(24,26)(27,32)(30,31)(33,38,39,47)(34,44,41,48)(35,43,46,36)(37,40,42,45)Polycyclic presentation:
2^9|a^2=ghi,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^c=fgi,e^d=egi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9388683);