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 | |
4496 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)(3,7)(4,8)(5,6)(9,10,14,15)(11,17,18,12)(13,21,22,16)(19,20,24,23)(25,26)(27,31)(28,33)(29,30)(32,38)(34,40)(35,39)(36,37);(2,6)(3,8)(10,15)(12,17)(16,21)(20,23)(25,27,30,37)(26,31,29,36)(28,35,38,34)(32,40,33,39);(1,3)(2,4)(5,8)(6,7)(10,15)(11,18)(13,22)(20,23)(25,28,30,38)(26,32,29,33)(27,34,37,35)(31,39,36,40);(1,4)(2,3)(5,7)(6,8)(9,11,14,18)(10,12,15,17)(13,19,22,24)(16,20,21,23)(25,28)(26,33)(27,35)(29,32)(30,38)(31,39)(34,37)(36,40);(1,2)(3,4)(5,6)(7,8)(9,12)(10,11)(13,20)(14,17)(15,18)(16,19)(21,24)(22,23)(25,27)(26,31)(28,34)(29,36)(30,37)(32,39)(33,40)(35,38);(1,4)(2,3)(5,7)(6,8)(9,13)(10,16)(11,19)(12,20)(14,22)(15,21)(17,23)(18,24)(25,29,30,26)(27,36,37,31)(28,32,38,33)(34,39,35,40)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=gh,d^2=hi,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, 8819465);