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 | |
4656 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10,14,12)(11,15,17,18)(13,16,21,20)(19,22,23,24)(25,26)(27,31)(28,35)(29,34)(30,39)(32,37)(33,38)(36,40);(3,7)(6,8)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,27,32,38)(26,33,37,31)(28,30,29,40)(34,39,35,36);(1,2)(3,6)(4,5)(7,8)(11,17)(15,18)(19,23)(22,24)(25,28,32,29)(26,34,37,35)(27,30,38,40)(31,36,33,39);(1,3,4,7)(2,6,5,8)(9,12,14,10)(11,18,17,15)(13,16,21,20)(19,22,23,24)(25,29,32,28)(26,35,37,34)(27,30,38,40)(31,36,33,39);(1,2)(3,6)(4,5)(7,8)(9,13)(10,16)(11,19)(12,20)(14,21)(15,22)(17,23)(18,24)(25,30)(26,36)(27,29)(28,38)(31,35)(32,40)(33,34)(37,39);(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,31)(26,27)(28,39)(29,36)(30,35)(32,33)(34,40)(37,38)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=g,d^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8198873);