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 | |
4622 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)(11,18)(17,22)(19,24)(25,26)(27,35)(28,39)(29,34)(30,31)(32,38)(33,37)(36,40);(1,2,5,6)(3,7,8,4)(9,10,14,15)(11,12,18,16)(13,17,21,22)(19,20,24,23)(25,27,31,38)(26,32,30,35)(28,29,37,36)(33,34,39,40);(1,2)(3,7)(4,8)(5,6)(11,18)(12,16)(13,21)(17,22)(26,30)(28,37)(29,36)(32,35);(2,6)(3,8)(9,11,14,18)(10,16,15,12)(13,19,21,24)(17,23,22,20)(25,28)(26,33)(27,36)(29,38)(30,39)(31,37)(32,40)(34,35);(1,3)(2,4)(5,8)(6,7)(9,12,14,16)(10,11,15,18)(13,20,21,23)(17,19,22,24)(25,29,31,36)(26,34,30,40)(27,37,38,28)(32,39,35,33);(1,4)(2,3)(5,7)(6,8)(9,13)(10,17)(11,19)(12,20)(14,21)(15,22)(16,23)(18,24)(25,30,31,26)(27,35,38,32)(28,39,37,33)(29,40,36,34)Polycyclic presentation:
2^9|b^2=gi,d^2=gh,e^2=g,f^2=h,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, 8968795);