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 | |
4381 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(5,8)(9,10,15,17)(11,18,12,16)(13,19,20,14)(21,24,23,22)(25,26)(27,29)(28,32)(30,31)(33,34)(35,38)(36,40)(37,39);(1,2,4,6)(3,5,7,8)(9,11,15,12)(10,16,17,18)(13,23,20,21)(14,22,19,24)(25,27)(26,29)(28,35)(30,34)(31,33)(32,38)(36,39)(37,40);(1,3)(2,5)(4,7)(6,8)(10,17)(14,19)(16,18)(22,24)(26,31)(27,34)(32,39)(35,40);(9,12,15,11)(10,18,17,16)(13,21,20,23)(14,24,19,22)(25,28)(26,32)(27,35)(29,38)(30,37)(31,39)(33,36)(34,40);(1,2)(3,5)(4,6)(7,8)(9,13,15,20)(10,19,17,14)(11,21,12,23)(16,22,18,24)(25,29,30,33)(26,27,31,34)(28,36,37,38)(32,40,39,35);(1,3)(2,5)(4,7)(6,8)(9,14,15,19)(10,20,17,13)(11,22,12,24)(16,21,18,23)(25,27)(26,33)(28,35)(29,31)(30,34)(32,36)(37,40)(38,39)Polycyclic presentation:
2^9|a^2=g,b^2=hi,d^2=g,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, 8807341);