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 | |
4516 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,17,18,16)(12,13,21,23)(19,20,22,24)(25,26)(27,29)(28,33)(30,39)(31,32)(34,38)(35,40)(36,37);(1,2)(3,8)(4,6)(5,7)(10,15)(12,21)(16,17)(19,22)(25,27,31,30)(26,29,32,39)(28,34,35,37)(33,38,40,36);(1,3)(2,4)(5,6)(7,8)(10,15)(11,18)(13,23)(19,22)(26,32)(27,30)(28,35)(36,38);(3,6)(4,8)(9,11)(10,16)(12,22)(13,20)(14,18)(15,17)(19,21)(23,24)(25,28)(26,33)(27,34)(29,38)(30,37)(31,35)(32,40)(36,39);(1,4,5,8)(2,6,7,3)(9,12)(10,13)(11,19)(14,21)(15,23)(16,24)(17,20)(18,22)(25,29)(26,27)(28,36)(30,32)(31,39)(33,37)(34,40)(35,38);(1,2)(3,8)(4,6)(5,7)(9,13)(10,12)(11,20)(14,23)(15,21)(16,22)(17,19)(18,24)(25,30,31,27)(26,29,32,39)(28,37,35,34)(33,38,40,36)Polycyclic presentation:
2^9|a^2=g,b^2=gh,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8821574);