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 | |
4728 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)(3,5)(4,8)(6,7)(9,10)(11,16)(12,13)(14,22)(15,17)(18,23)(19,21)(20,24)(25,26,31,34)(27,32,35,39)(28,38,37,30)(29,33,36,40);(2,7)(3,8)(9,11,15,12)(10,16,17,13)(14,19,23,20)(18,24,22,21)(25,27,31,35)(26,32,34,39)(28,36,37,29)(30,33,38,40);(1,3,6,8)(2,4,7,5)(10,17)(11,12)(14,23)(21,24)(27,35)(28,37)(30,38)(32,39);(1,4,6,5)(2,3,7,8)(9,12,15,11)(10,16,17,13)(14,20,23,19)(18,24,22,21)(25,28)(26,30)(27,36)(29,35)(31,37)(32,33)(34,38)(39,40);(1,2)(3,4)(5,8)(6,7)(9,13,15,16)(10,12,17,11)(14,21,23,24)(18,20,22,19)(25,29,31,36)(26,33,34,40)(27,37,35,28)(30,39,38,32);(1,5,6,4)(2,8,7,3)(9,14)(10,18)(11,19)(12,20)(13,21)(15,23)(16,24)(17,22)(25,30)(26,28)(27,33)(29,39)(31,38)(32,36)(34,37)(35,40)Polycyclic presentation:
2^9|a^2=hi,b^2=gi,c^2=g,d^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
| maximal normal subgroup of | #72219281, #72219285 |
| maximal quotient of | #72219270 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8839469);