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 | |
4378 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)(3,7)(4,6)(5,8)(10,14)(11,17)(16,19)(18,20)(27,33)(31,35)(34,39)(38,40);(1,2,4,6)(3,5,8,7)(9,10,13,14)(11,15,17,22)(12,16,21,19)(18,23,20,24)(25,26,30,29)(27,31,33,35)(28,36,37,32)(34,40,39,38);(1,2)(3,5)(4,6)(7,8)(9,10,13,14)(11,15,17,22)(12,16,21,19)(18,23,20,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,19)(13,14)(16,21)(17,22)(18,24)(20,23)(25,28,30,37)(26,32,29,36)(27,34,33,39)(31,38,35,40);(9,11)(10,15)(12,20)(13,17)(14,22)(16,24)(18,21)(19,23)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39);(9,12,13,21)(10,16,14,19)(11,18,17,20)(15,23,22,24)(25,29,30,26)(27,35,33,31)(28,36,37,32)(34,40,39,38)Polycyclic presentation:
2^9|b^2=hi,c^2=gi,d^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8627501);