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 | |
4460 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)(3,5)(4,8)(6,7)(9,10,14,17)(11,18,12,15)(13,16,21,23)(19,24,20,22);(2,7)(3,8)(9,11,14,12)(10,15,17,18)(13,19,21,20)(16,22,23,24)(25,26,30,31)(27,29,33,38)(28,32,37,39)(34,36,35,40);(1,3,6,8)(2,4,7,5)(13,21)(16,23)(19,20)(22,24)(26,31)(29,38)(32,39)(36,40);(1,4,6,5)(2,3,7,8)(9,10)(11,15)(12,18)(13,16)(14,17)(19,22)(20,24)(21,23)(25,27,30,33)(26,29,31,38)(28,35,37,34)(32,40,39,36);(1,2)(3,4)(5,8)(6,7)(9,12,14,11)(10,15,17,18)(13,20,21,19)(16,22,23,24)(25,28,30,37)(26,32,31,39)(27,34,33,35)(29,36,38,40);(1,5,6,4)(2,8,7,3)(9,13,14,21)(10,16,17,23)(11,19,12,20)(15,22,18,24)(25,29)(26,33)(27,31)(28,36)(30,38)(32,35)(34,39)(37,40)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=h,d^2=i,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8809687);