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 | |
4351 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)(4,5)(6,8)(9,10)(11,14)(12,19)(13,15)(16,20)(17,23)(18,21)(22,24)(25,26)(27,34)(28,36)(29,32)(30,31)(33,38)(35,39)(37,40);(3,8)(6,7)(9,11,13,18)(10,14,15,21)(12,17,20,24)(16,22,19,23)(25,26,31,30)(27,32,29,34)(28,33,38,36)(35,39,37,40);(1,3)(2,6)(4,7)(5,8)(10,15)(14,21)(16,19)(22,23)(25,27,31,29)(26,32,30,34)(28,37,38,35)(33,40,36,39);(3,8)(6,7)(9,10)(11,14)(12,16)(13,15)(17,22)(18,21)(19,20)(23,24)(25,28,31,38)(26,33,30,36)(27,35,29,37)(32,39,34,40);(1,3)(2,7)(4,6)(5,8)(9,11)(10,14)(12,17)(13,18)(15,21)(16,22)(19,23)(20,24)(25,29,31,27)(26,32,30,34)(28,35,38,37)(33,40,36,39);(1,4,5,2)(3,6,8,7)(9,12,13,20)(10,16,15,19)(11,17,18,24)(14,22,21,23)(25,30,31,26)(27,32,29,34)(28,36,38,33)(35,39,37,40)Polycyclic presentation:
2^9|b^2=hi,c^2=g,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
| maximal normal subgroup of | #72219279 |
| maximal quotient of | #72219273 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9347766);