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 | |
4261 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,17,18,12)(13,21,22,16)(19,20,24,23)(25,26)(27,28)(29,40)(30,35)(31,36)(32,37)(33,39)(34,38);(1,2)(3,4)(5,6)(7,8)(9,11,14,18)(10,12,15,17)(13,19,22,24)(16,20,21,23)(25,27)(26,33)(28,37)(29,35)(30,34)(31,38)(32,39)(36,40);(2,6)(3,7)(10,15)(11,18)(16,21)(19,24)(25,28,32,33)(26,27,37,39)(29,36,38,30)(31,34,35,40);(3,7)(4,8)(9,10)(11,12)(13,16)(14,15)(17,18)(19,20)(21,22)(23,24)(25,29)(26,34)(27,35)(28,36)(30,33)(31,39)(32,38)(37,40);(1,3)(2,4)(5,7)(6,8)(9,12)(10,11)(13,20)(14,17)(15,18)(16,19)(21,24)(22,23)(25,30,32,36)(26,35,37,31)(27,34,39,40)(28,38,33,29);(1,4)(2,3)(5,8)(6,7)(9,13,14,22)(10,16,15,21)(11,19,18,24)(12,20,17,23)(25,31,32,35)(26,36,37,30)(27,38,39,29)(28,34,33,40)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,e^d=egi,f^d=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9532658);