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 | |
4512 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)(4,5)(6,8)(9,10)(11,20)(12,14)(13,23)(15,19)(16,24)(17,22)(18,21)(25,26,29,32)(27,30,35,33)(28,31,37,40)(34,38,36,39);(2,5)(6,7)(9,11)(10,16)(12,21)(13,22)(14,23)(15,24)(17,18)(19,20)(25,26,29,32)(27,33,35,30)(28,31,37,40)(34,39,36,38);(1,2)(3,6)(4,5)(7,8)(25,26)(27,30)(28,31)(29,32)(33,35)(34,38)(36,39)(37,40);(1,3,4,8)(2,6,5,7)(9,12,15,22)(10,17,19,14)(11,21,24,13)(16,18,20,23)(25,27)(26,30)(28,36)(29,35)(31,39)(32,33)(34,37)(38,40);(9,13,15,21)(10,18,19,23)(11,22,24,12)(14,16,17,20);(9,14,15,17)(10,12,19,22)(11,23,24,18)(13,16,21,20)(25,28,29,37)(26,31,32,40)(27,34,35,36)(30,38,33,39)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=g,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8366029);