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 | |
4418 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,12)(13,17)(14,15)(16,19)(18,20)(21,22)(23,24)(25,26,30,32)(27,33,36,31)(28,37,39,29)(34,40,38,35);(1,2,5,4)(3,7,8,6)(10,15)(12,16)(17,22)(20,23)(25,27,30,36)(26,31,32,33)(28,38,39,34)(29,35,37,40);(2,4)(6,7)(13,21)(17,22)(18,24)(20,23)(26,32)(29,37)(31,33)(35,40);(3,8)(6,7)(9,11,14,19)(10,16,15,12)(13,18,21,24)(17,23,22,20)(25,27)(26,31)(28,38)(29,35)(30,36)(32,33)(34,39)(37,40);(1,3)(2,6)(4,7)(5,8)(9,12,14,16)(10,11,15,19)(13,20,21,23)(17,18,22,24)(25,28)(26,29)(27,34)(30,39)(31,40)(32,37)(33,35)(36,38);(1,4,5,2)(3,7,8,6)(9,13,14,21)(10,17,15,22)(11,18,19,24)(12,20,16,23)(25,29)(26,28)(27,35)(30,37)(31,38)(32,39)(33,34)(36,40)Polycyclic presentation:
2^9|a^2=h,b^2=hi,d^2=gh,e^2=gh,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8282719);