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 | |
4478 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10)(11,16)(12,18)(13,23)(14,24)(15,17)(19,22)(20,21)(25,26,30,34)(27,35,37,31)(28,32,38,40)(29,39,36,33);(3,7)(6,8)(9,11)(10,16)(12,15)(13,19)(14,20)(17,18)(21,24)(22,23)(25,27,30,37)(26,31,34,35)(28,36,38,29)(32,39,40,33);(1,2)(3,6)(4,5)(7,8)(10,17)(11,12)(13,24)(20,22)(25,28,30,38)(26,32,34,40)(27,36,37,29)(31,39,35,33);(1,3)(2,6)(4,7)(5,8)(9,12,15,11)(10,18,17,16)(13,21,24,19)(14,22,23,20)(25,29,30,36)(26,33,34,39)(27,28,37,38)(31,32,35,40);(1,2)(3,6)(4,5)(7,8)(9,13)(10,14)(11,19)(12,21)(15,24)(16,20)(17,23)(18,22)(25,28)(26,32)(27,36)(29,37)(30,38)(31,39)(33,35)(34,40);(9,14)(10,13)(11,20)(12,22)(15,23)(16,19)(17,24)(18,21)(25,26)(27,31)(28,32)(29,33)(30,34)(35,37)(36,39)(38,40)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=g,d^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8034851);