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 | |
4526 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,15)(11,17)(16,20)(19,22)(25,26)(27,30)(28,36)(29,32)(31,37)(33,35)(34,39)(38,40);(1,2,5,6)(3,7,8,4)(9,10,14,15)(11,18,17,13)(12,20,21,16)(19,23,22,24)(25,27)(26,30)(28,34)(29,35)(31,38)(32,33)(36,39)(37,40);(1,2)(3,7)(4,8)(5,6)(11,17)(12,21)(13,18)(16,20)(25,26,29,32)(27,33,35,30)(28,36,37,31)(34,38,40,39);(2,6)(3,8)(9,11)(10,13)(12,22)(14,17)(15,18)(16,24)(19,21)(20,23)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,37)(38,40);(1,3)(2,4)(5,8)(6,7)(9,12,14,21)(10,16,15,20)(11,19,17,22)(13,23,18,24)(25,27)(26,30)(28,34)(29,35)(31,38)(32,33)(36,39)(37,40);(1,4)(2,3)(5,7)(6,8)(9,13)(10,11)(12,23)(14,18)(15,17)(16,19)(20,22)(21,24)(25,28)(26,31)(27,34)(29,37)(30,38)(32,36)(33,39)(35,40)Polycyclic presentation:
2^9|b^2=ghi,c^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8968329);