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 |
| prime radical | #44237 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4336 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10,14,15)(11,16,17,12)(13,18,21,22)(19,23,24,20)(25,26,31,35)(27,34,29,32)(28,33,39,30)(36,37,38,40);(1,2)(3,6)(4,7)(5,8)(9,11)(10,12)(13,19)(14,17)(15,16)(18,20)(21,24)(22,23)(25,27,31,29)(26,32,35,34)(28,36,39,38)(30,37,33,40);(10,15)(11,17)(13,21)(20,23)(25,28,31,39)(26,30,35,33)(27,36,29,38)(32,37,34,40);(1,3,5,4)(2,6,8,7)(9,11)(10,16)(12,15)(13,19)(14,17)(18,23)(20,22)(21,24)(25,28)(26,33)(27,36)(29,38)(30,35)(31,39)(32,40)(34,37);(9,12,14,16)(10,17,15,11)(13,20,21,23)(18,24,22,19)(25,29,31,27)(26,34,35,32)(28,38,39,36)(30,40,33,37);(1,4,5,3)(2,7,8,6)(9,13)(10,18)(11,19)(12,20)(14,21)(15,22)(16,23)(17,24)(25,30,31,33)(26,28,35,39)(27,37,29,40)(32,36,34,38)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=hi,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7689914);