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 | |
4715 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)(11,12,17,20)(13,16,23,24)(18,19,21,22)(25,26)(27,30)(28,32)(29,33)(31,34)(35,38)(36,37)(39,40);(3,7)(6,8)(10,15)(11,17)(16,24)(18,21)(25,27)(26,30)(28,35)(29,34)(31,33)(32,38)(36,39)(37,40);(1,2)(3,6)(4,5)(7,8)(10,15)(11,17)(16,24)(18,21)(27,34)(30,31)(35,40)(38,39);(1,3)(2,6)(4,7)(5,8)(9,11)(10,12)(13,21)(14,17)(15,20)(16,22)(18,23)(19,24)(25,27)(26,31)(28,35)(29,34)(30,33)(32,39)(36,38)(37,40);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,22)(14,20)(15,17)(16,21)(18,24)(19,23)(25,28)(26,32)(27,35)(29,37)(30,38)(31,39)(33,36)(34,40);(9,13,14,23)(10,16,15,24)(11,18,17,21)(12,19,20,22)(25,26)(27,30)(28,36)(29,33)(31,34)(32,37)(35,39)(38,40)Polycyclic presentation:
2^9|a^2=g,f^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, 8633943);