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 | |
4388 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,20)(12,17)(13,24)(14,19)(15,22)(16,23)(18,21)(25,26)(27,31)(28,37)(29,33)(30,34)(32,39)(35,36)(38,40);(3,7)(5,8)(9,11)(10,15)(12,13)(14,22)(16,24)(17,18)(19,20)(21,23)(25,26)(27,35)(28,32)(29,33)(30,34)(31,36)(37,39)(38,40);(1,2)(3,5)(4,6)(7,8)(9,12,14,23)(10,16,19,17)(11,13,22,21)(15,24,20,18)(25,27,30,36)(26,31,34,35)(28,38,39,29)(32,33,37,40);(1,3)(2,5)(4,7)(6,8)(9,12)(10,17)(11,21)(13,22)(14,23)(15,24)(16,19)(18,20)(25,28,30,39)(26,32,34,37)(27,29,36,38)(31,40,35,33);(1,2)(3,5)(4,6)(7,8)(9,13,14,21)(10,18,19,24)(11,12,22,23)(15,17,20,16)(25,26)(27,31)(28,32)(29,40)(30,34)(33,38)(35,36)(37,39);(9,10)(11,15)(12,16)(13,24)(14,19)(17,23)(18,21)(20,22)(25,29)(26,33)(27,28)(30,38)(31,37)(32,35)(34,40)(36,39)Polycyclic presentation:
2^9|a^2=gh,c^2=gi,d^2=i,e^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8650652);