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 | |
4582 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,4)(6,7)(9,10)(11,12)(13,21)(14,15)(16,18)(17,22)(19,23)(20,24)(25,26,31,29)(27,32,34,30)(28,33,36,38)(35,39,40,37);(1,2,5,8)(3,6,4,7)(10,15)(12,16)(17,21)(20,23)(25,27,31,34)(26,32,29,30)(28,35,36,40)(33,39,38,37);(1,3,5,4)(2,6,8,7)(10,15)(11,18)(13,22)(20,23)(27,34)(28,36)(30,32)(33,38);(9,11,14,18)(10,16,15,12)(13,19,22,24)(17,23,21,20)(25,28)(26,33)(27,35)(29,38)(30,37)(31,36)(32,39)(34,40);(1,2)(3,6)(4,7)(5,8)(9,12,14,16)(10,11,15,18)(13,20,22,23)(17,19,21,24)(25,29,31,26)(27,30,34,32)(28,33,36,38)(35,39,40,37);(1,4,5,3)(2,7,8,6)(9,13)(10,17)(11,19)(12,20)(14,22)(15,21)(16,23)(18,24)(25,30)(26,27)(28,37)(29,34)(31,32)(33,35)(36,39)(38,40)Polycyclic presentation:
2^9|a^2=i,b^2=ghi,c^2=gh,d^2=g,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
| maximal normal subgroup of | #72219279, #72219285 |
| maximal quotient of | #72219275 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8827918);