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 | |
4544 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10,14,15)(11,16,19,22)(12,17,21,24)(13,20,23,18)(25,26,30,34)(27,31,35,32)(28,29,33,39)(36,37,38,40);(3,7)(6,8)(10,15)(12,21)(16,22)(18,20)(25,27)(26,31)(28,36)(29,37)(30,35)(32,34)(33,38)(39,40);(1,2)(3,6)(4,5)(7,8)(10,15)(11,19)(17,24)(18,20)(27,35)(31,32)(36,38)(37,40);(9,11)(10,16)(12,18)(13,17)(14,19)(15,22)(20,21)(23,24)(25,27)(26,32)(28,38)(29,37)(30,35)(31,34)(33,36)(39,40);(1,3)(2,6)(4,7)(5,8)(9,12,14,21)(10,17,15,24)(11,20,19,18)(13,16,23,22)(25,28)(26,29)(27,36)(30,33)(31,37)(32,40)(34,39)(35,38);(1,2)(3,6)(4,5)(7,8)(9,13,14,23)(10,18,15,20)(11,17,19,24)(12,22,21,16)(25,29)(26,33)(27,37)(28,34)(30,39)(31,38)(32,36)(35,40)Polycyclic presentation:
2^9|a^2=gi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9077654);