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 | |
4452 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10,15,13)(11,16,19,21)(12,17,22,23)(14,20,24,18)(25,26,31,34)(27,32,35,28)(29,30,33,40)(36,39,38,37);(2,3)(6,7)(9,11)(10,16)(12,18)(13,21)(14,17)(15,19)(20,22)(23,24)(25,27)(26,32)(28,34)(29,38)(30,37)(31,35)(33,36)(39,40);(1,3,5,2)(4,7,8,6)(10,13)(11,19)(12,22)(14,24)(27,35)(28,32)(29,33)(30,40);(1,4)(2,6)(3,7)(5,8)(9,12)(10,17)(11,20)(13,23)(14,21)(15,22)(16,24)(18,19)(25,28)(26,27)(29,39)(30,38)(31,32)(33,37)(34,35)(36,40);(9,13,15,10)(11,21,19,16)(12,17,22,23)(14,20,24,18)(25,29,31,33)(26,30,34,40)(27,36,35,38)(28,37,32,39);(9,14,15,24)(10,18,13,20)(11,17,19,23)(12,21,22,16)(25,30)(26,33)(27,37)(28,38)(29,34)(31,40)(32,36)(35,39)Polycyclic presentation:
2^9|a^2=i,c^2=g,e^2=gi,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, 8970473);