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 | |
4674 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,4)(5,6)(7,8)(9,10,13,15)(11,17,19,14)(12,16,21,23)(18,24,20,22)(25,26)(27,32)(28,33)(29,38)(30,31)(34,35)(36,39)(37,40);(2,6)(4,7)(9,11)(10,14)(12,20)(13,19)(15,17)(16,24)(18,21)(22,23)(25,26)(27,32)(28,29)(30,31)(33,38)(34,40)(35,37)(36,39);(1,2,5,6)(3,7,8,4)(10,15)(14,17)(16,23)(22,24)(26,31)(29,33)(32,39)(35,40);(1,3)(2,7)(4,6)(5,8)(9,11)(10,14)(12,20)(13,19)(15,17)(16,24)(18,21)(22,23)(25,27,30,36)(26,32,31,39)(28,37,38,34)(29,35,33,40);(1,4,5,7)(2,3,6,8)(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(25,28)(26,33)(27,34)(29,31)(30,38)(32,35)(36,37)(39,40);(1,3)(2,7)(4,6)(5,8)(9,10)(11,14)(12,16)(13,15)(17,19)(18,22)(20,24)(21,23)(25,29,30,33)(26,28,31,38)(27,35,36,40)(32,37,39,34)Polycyclic presentation:
2^9|a^2=hi,c^2=ghi,d^2=gh,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8806136);