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 | |
4338 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,12)(11,18,13,16)(14,20,22,17)(19,23,21,24)(26,32)(27,30)(28,34)(37,39);(1,2)(3,6)(4,5)(7,8)(9,10,15,12)(11,16,13,18)(14,17,22,20)(19,23,21,24)(25,26)(27,35)(28,29)(30,33)(31,32)(34,40)(36,37)(38,39);(2,5)(6,8)(9,11,15,13)(10,16,12,18)(14,19,22,21)(17,23,20,24)(25,27,31,30)(26,33,32,35)(28,38,34,36)(29,37,40,39);(3,7)(6,8)(9,12,15,10)(11,16,13,18)(14,20,22,17)(19,23,21,24)(25,28,31,34)(26,29,32,40)(27,36,30,38)(33,39,35,37);(1,3,4,7)(2,6,5,8)(9,13,15,11)(10,16,12,18)(14,21,22,19)(17,23,20,24)(25,29)(26,34)(27,37)(28,32)(30,39)(31,40)(33,36)(35,38);(1,2)(3,6)(4,5)(7,8)(9,14)(10,17)(11,19)(12,20)(13,21)(15,22)(16,23)(18,24)(25,30,31,27)(26,33,32,35)(28,38,34,36)(29,39,40,37)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=h,d^2=h,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9353496);