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 | |
4478 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10,14,16)(11,17,20,15)(12,18,22,24)(13,19,23,21)(25,26)(27,30)(28,32)(29,33)(31,34)(35,38)(36,37)(39,40);(1,3,5,8)(2,6,4,7)(9,11)(10,15)(12,19)(13,18)(14,20)(16,17)(21,22)(23,24)(25,27)(26,30)(28,35)(29,34)(31,33)(32,38)(36,39)(37,40);(10,16)(13,23)(15,17)(18,24)(27,34)(30,31)(35,40)(38,39);(9,11)(10,17)(12,21)(13,18)(14,20)(15,16)(19,22)(23,24)(25,26)(27,30)(28,36)(29,33)(31,34)(32,37)(35,39)(38,40);(1,2,5,4)(3,6,8,7)(9,12,14,22)(10,18,16,24)(11,19,20,21)(13,17,23,15)(25,27)(26,31)(28,35)(29,34)(30,33)(32,39)(36,38)(37,40);(1,4,5,2)(3,7,8,6)(9,13,14,23)(10,19,16,21)(11,18,20,24)(12,17,22,15)(25,28)(26,32)(27,35)(29,37)(30,38)(31,39)(33,36)(34,40)Polycyclic presentation:
2^9|a^2=g,b^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7976880);