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 | |
4776 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)(3,6)(4,7)(5,8)(9,10)(11,17)(12,19)(13,16)(14,15)(18,24)(20,21)(22,23)(26,30)(27,33)(28,35)(39,40);(1,3,5,7)(2,6,8,4)(10,15)(12,17)(16,23)(20,24)(25,26,29,30)(27,31,33,37)(28,32,35,38)(34,39,36,40);(1,3)(2,4)(5,7)(6,8)(25,27)(26,31)(28,34)(29,33)(30,37)(32,39)(35,36)(38,40);(1,4,5,6)(2,7,8,3)(9,11,14,19)(10,12,15,17)(13,21,22,18)(16,24,23,20)(25,27)(26,31)(28,36)(29,33)(30,37)(32,40)(34,35)(38,39);(1,3)(2,4)(5,7)(6,8)(9,12)(10,11)(13,20)(14,17)(15,19)(16,18)(21,23)(22,24)(25,26)(27,31)(28,32)(29,30)(33,37)(34,39)(35,38)(36,40);(1,4)(2,3)(5,6)(7,8)(9,13,14,22)(10,16,15,23)(11,18,19,21)(12,20,17,24)(25,28,29,35)(26,32,30,38)(27,34,33,36)(31,39,37,40)Polycyclic presentation:
2^9|b^2=h,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10315730);