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 | |
4624 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)(9,10,15,12)(11,16,22,19)(13,23,21,17)(14,18,20,24)(25,26,31,30)(27,32,38,37)(28,34,29,33)(35,40,36,39);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,19)(13,20)(14,21)(15,22)(17,18)(23,24)(25,27,31,38)(26,32,30,37)(28,35,29,36)(33,39,34,40);(9,10,15,12)(11,19,22,16)(13,17,21,23)(14,18,20,24)(25,28,31,29)(26,33,30,34)(27,35,38,36)(32,39,37,40);(2,6)(5,7)(9,12,15,10)(11,19,22,16)(13,23,21,17)(14,18,20,24)(25,27)(26,32)(28,35)(29,36)(30,37)(31,38)(33,39)(34,40);(1,3)(2,5)(4,8)(6,7)(9,13,15,21)(10,17,12,23)(11,20,22,14)(16,18,19,24)(25,29,31,28)(26,34,30,33)(27,36,38,35)(32,40,37,39);(1,2)(3,5)(4,6)(7,8)(9,14,15,20)(10,18,12,24)(11,21,22,13)(16,17,19,23)(25,30,31,26)(27,37,38,32)(28,34,29,33)(35,40,36,39)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=hi,d^2=i,e^2=hi,f^2=hi,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, 8378388);