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 | |
4664 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)(4,5)(6,8)(9,10)(11,17)(12,13)(14,18)(15,21)(16,22)(19,23)(20,24)(26,32)(27,28)(29,37)(38,39);(2,5)(6,7)(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24)(25,26)(27,33)(28,30)(29,34)(31,32)(35,38)(36,39)(37,40);(1,2)(3,6)(4,5)(7,8)(25,27,31,28)(26,33,32,30)(29,35,37,36)(34,38,40,39);(1,3)(2,6)(4,8)(5,7)(9,12,14,22)(10,16,18,13)(11,19,21,24)(15,23,17,20)(25,28,31,27)(26,33,32,30)(29,36,37,35)(34,38,40,39);(9,11)(10,17)(12,19)(13,23)(14,21)(15,18)(16,20)(22,24)(25,29)(26,34)(27,35)(28,36)(30,39)(31,37)(32,40)(33,38);(9,13,14,16)(10,12,18,22)(11,20,21,23)(15,19,17,24)(25,30,31,33)(26,28,32,27)(29,38,37,39)(34,35,40,36)Polycyclic presentation:
2^9|c^2=h,d^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8587193);