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 | |
4473 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,7)(9,10,15,17)(11,16,21,24)(12,18,20,14)(13,23,22,19)(25,26)(27,33)(28,36)(29,30)(31,35)(32,37)(34,38)(39,40);(1,2,5,4)(3,7,8,6)(9,11)(10,16)(12,22)(13,20)(14,23)(15,21)(17,24)(18,19)(25,27,30,35)(26,31,29,33)(28,34,37,40)(32,38,36,39);(1,3,5,8)(2,6,4,7)(10,17)(11,21)(12,20)(19,23)(28,37)(32,36)(34,40)(38,39);(1,4,5,2)(3,7,8,6)(9,12,15,20)(10,18,17,14)(11,22,21,13)(16,19,24,23)(25,28,30,37)(26,32,29,36)(27,34,35,40)(31,38,33,39);(9,13,15,22)(10,19,17,23)(11,20,21,12)(14,16,18,24)(25,29,30,26)(27,31,35,33)(28,36,37,32)(34,38,40,39);(9,14,15,18)(10,20,17,12)(11,19,21,23)(13,16,22,24)(25,27)(26,33)(28,34)(29,31)(30,35)(32,39)(36,38)(37,40)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=hi,d^2=gh,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8664321);