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 | |
4495 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)(3,7)(4,6)(5,8)(9,10)(11,16)(12,18)(13,21)(14,20)(15,17)(19,24)(22,23);(1,2,4,6)(3,5,8,7)(9,11,15,14)(10,16,17,20)(12,19,23,21)(13,18,24,22)(25,26,30,31)(27,29,33,38)(28,35,37,32)(34,40,39,36);(1,2)(3,5)(4,6)(7,8)(10,17)(11,14)(18,22)(19,21)(26,31)(29,38)(32,35)(36,40);(1,3)(2,5)(4,8)(6,7)(9,12,15,23)(10,18,17,22)(11,21,14,19)(13,20,24,16)(25,27,30,33)(26,29,31,38)(28,34,37,39)(32,36,35,40);(9,13,15,24)(10,19,17,21)(11,22,14,18)(12,20,23,16)(25,28,30,37)(26,32,31,35)(27,34,33,39)(29,36,38,40);(9,14,15,11)(10,20,17,16)(12,19,23,21)(13,18,24,22)(25,29)(26,33)(27,31)(28,36)(30,38)(32,39)(34,35)(37,40)Polycyclic presentation:
2^9|b^2=i,d^2=gi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8135881);