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 | |
4561 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)(11,13)(12,16)(14,15)(17,21)(18,20)(19,22)(23,24)(25,26,31,33)(27,32,35,30)(28,29,34,40)(36,37,39,38);(2,6)(4,7)(12,19)(16,22)(18,23)(20,24)(25,27)(26,32)(28,36)(29,37)(30,33)(31,35)(34,39)(38,40);(10,15)(11,17)(16,22)(18,23)(26,33)(27,35)(28,34)(37,38);(1,3)(2,7)(4,6)(5,8)(9,11,14,17)(10,13,15,21)(12,18,19,23)(16,20,22,24)(25,28,31,34)(26,29,33,40)(27,36,35,39)(30,38,32,37);(1,4,5,7)(2,3,6,8)(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,29)(26,34)(27,37)(28,33)(30,36)(31,40)(32,39)(35,38);(1,3)(2,4)(5,8)(6,7)(9,13)(10,17)(11,15)(12,20)(14,21)(16,23)(18,22)(19,24)(25,30)(26,35)(27,33)(28,38)(29,39)(31,32)(34,37)(36,40)Polycyclic presentation:
2^9|a^2=i,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9690264);