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 | |
4538 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)(11,17,18,12)(13,21,23,16)(19,20,24,22)(25,26)(27,35)(28,30)(29,32)(31,34)(33,39)(36,40)(37,38);(1,2)(3,6)(4,7)(5,8)(10,15)(12,17)(16,21)(20,22)(25,27,31,29)(26,32,34,35)(28,36,39,38)(30,37,33,40);(1,3)(2,5)(4,8)(6,7)(10,15)(11,18)(16,21)(19,24);(9,11)(10,12)(13,19)(14,18)(15,17)(16,20)(21,22)(23,24)(25,28)(26,33)(27,36)(29,38)(30,34)(31,39)(32,40)(35,37);(1,3)(2,6)(4,8)(5,7)(9,12)(10,11)(13,22)(14,17)(15,18)(16,24)(19,21)(20,23)(25,29,31,27)(26,32,34,35)(28,38,39,36)(30,37,33,40);(1,2)(3,5)(4,7)(6,8)(9,13,14,23)(10,16,15,21)(11,19,18,24)(12,20,17,22)(25,30,31,33)(26,28,34,39)(27,37,29,40)(32,36,35,38)Polycyclic presentation:
2^9|a^2=h,b^2=gh,e^2=gh,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^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9576560);