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 | |
4635 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)(11,13)(12,20)(14,16)(15,18)(17,22)(19,24)(21,23)(25,26)(27,30)(28,36)(29,32)(31,37)(33,35)(34,39)(38,40);(1,2,5,6)(3,7,8,4)(9,11,14,18)(10,15,16,13)(12,21,22,19)(17,23,20,24)(25,27)(26,30)(28,34)(29,35)(31,38)(32,33)(36,39)(37,40);(1,2)(3,7)(4,8)(5,6)(10,16)(12,22)(13,15)(19,21)(25,26,29,32)(27,33,35,30)(28,36,37,31)(34,38,40,39);(2,6)(3,8)(9,12)(10,17)(11,19)(13,23)(14,22)(15,24)(16,20)(18,21)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,37)(38,40);(1,3)(2,4)(5,8)(6,7)(9,10)(11,15)(12,20)(13,18)(14,16)(17,22)(19,23)(21,24)(25,27)(26,30)(28,34)(29,35)(31,38)(32,33)(36,39)(37,40);(1,4)(2,3)(5,7)(6,8)(9,13,14,15)(10,18,16,11)(12,23,22,24)(17,21,20,19)(25,28)(26,31)(27,34)(29,37)(30,38)(32,36)(33,39)(35,40)Polycyclic presentation:
2^9|b^2=hi,c^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8968341);