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 | |
4695 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)(11,18)(12,13)(14,15)(16,19)(17,22)(20,21)(23,24)(25,26,30,29)(27,31,33,35)(28,32,36,37)(34,38,39,40);(1,2,4,6)(3,5,8,7)(10,15)(11,19)(12,22)(21,23)(27,33)(31,35)(34,39)(38,40);(1,2)(3,5)(4,6)(7,8)(10,15)(12,22)(16,18)(20,24)(25,27,30,33)(26,31,29,35)(28,34,36,39)(32,38,37,40);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,20)(13,23)(14,19)(15,18)(17,21)(22,24)(25,28)(26,32)(27,34)(29,37)(30,36)(31,38)(33,39)(35,40);(9,12)(10,17)(11,20)(13,15)(14,22)(16,21)(18,23)(19,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,39)(37,40);(9,13,14,17)(10,12,15,22)(11,21,19,23)(16,24,18,20)(25,29,30,26)(27,35,33,31)(28,32,36,37)(34,38,39,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8546973);