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 | |
4398 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(25,26)(27,30)(28,36)(29,32)(31,37)(33,35)(34,39)(38,40);(1,3,5,8)(2,6,4,7)(9,10,14,15)(11,13,17,22)(12,19,21,16)(18,24,23,20)(25,26)(27,33)(28,36)(29,32)(30,35)(31,37)(34,38)(39,40);(2,4)(6,7)(10,15)(13,22)(16,19)(20,24)(25,26,29,32)(27,30,35,33)(28,31,37,36)(34,38,40,39);(1,3)(2,6)(4,7)(5,8)(9,11,14,17)(10,13,15,22)(12,18,21,23)(16,20,19,24)(25,27)(26,30)(28,34)(29,35)(31,38)(32,33)(36,39)(37,40);(1,3)(2,6)(4,7)(5,8)(9,12,14,21)(10,16,15,19)(11,18,17,23)(13,20,22,24)(25,28)(26,31)(27,34)(29,37)(30,38)(32,36)(33,39)(35,40);(1,4,5,2)(3,7,8,6)(9,13)(10,17)(11,15)(12,20)(14,22)(16,23)(18,19)(21,24)(25,26)(27,30)(28,31)(29,32)(33,35)(34,38)(36,37)(39,40)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=gh,d^2=ghi,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8135790);