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 | |
3973 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)(3,8)(4,6)(5,14)(7,9)(10,13)(11,16)(12,15)(17,18,23,21)(19,20,26,24)(22,29,30,25)(27,31,32,28)(34,38)(36,43)(39,44)(42,47);(1,3)(2,8)(4,13)(5,11)(6,10)(7,12)(9,15)(14,16)(18,21)(19,26)(25,29)(27,32)(35,41)(39,44)(42,47)(46,48);(2,9)(3,12)(4,11)(10,14)(17,19,23,26)(18,24,21,20)(22,27,30,32)(25,31,29,28)(33,34,37,38)(35,39,41,44)(36,40,43,45)(42,46,47,48);(1,4,7,11)(2,6,9,16)(3,13,12,5)(8,10,15,14)(17,20)(18,19)(21,26)(22,28)(23,24)(25,27)(29,32)(30,31);(1,5,7,13)(2,10,9,14)(3,11,12,4)(6,15,16,8)(17,21,23,18)(19,20,26,24)(22,29,30,25)(27,28,32,31)(33,35,37,41)(34,39,38,44)(36,42,43,47)(40,46,45,48);(1,6)(2,11)(3,14)(4,9)(5,15)(7,16)(8,13)(10,12)(17,22,23,30)(18,25,21,29)(19,27,26,32)(20,28,24,31)(33,36)(34,40)(35,42)(37,43)(38,45)(39,46)(41,47)(44,48)Polycyclic presentation:
2^9|a^2=g,c^2=ghi,d^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8662235);