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 | |
4577 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,11)(12,15,19,17)(13,18,21,16)(20,23,24,22)(25,26,31,30)(27,35,37,32)(28,33,36,38)(29,34,39,40);(1,2,5,6)(3,7,8,4)(12,19)(15,17)(20,24)(22,23)(25,27,31,37)(26,32,30,35)(28,29,36,39)(33,40,38,34);(1,3)(2,4)(5,8)(6,7)(10,11)(15,17)(16,18)(22,23)(25,28,31,36)(26,33,30,38)(27,29,37,39)(32,40,35,34);(2,6)(4,7)(9,11,14,10)(12,17,19,15)(13,18,21,16)(20,23,24,22)(25,27)(26,32)(28,29)(30,35)(31,37)(33,40)(34,38)(36,39);(1,4)(2,3)(5,7)(6,8)(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24)(25,29)(26,34)(27,36)(28,37)(30,40)(31,39)(32,33)(35,38);(1,3)(2,7)(4,6)(5,8)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,30,31,26)(27,35,37,32)(28,38,36,33)(29,34,39,40)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=g,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10133129);