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 | |
4577 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)(9,10,14,16)(11,17,18,15)(12,21,23,13)(19,20,22,24)(25,26)(27,37)(28,33)(29,34)(30,35)(31,36)(32,38)(39,40);(1,2,4,5)(3,6,7,8)(9,11)(10,15)(12,19)(13,20)(14,18)(16,17)(21,24)(22,23)(25,27)(26,32)(28,30)(29,35)(31,38)(33,40)(34,39)(36,37);(10,16)(11,18)(13,21)(19,22)(27,38)(30,39)(32,37)(35,40);(9,11)(10,17)(12,22)(13,20)(14,18)(15,16)(19,23)(21,24)(25,28)(26,33)(27,30)(29,36)(31,34)(32,40)(35,37)(38,39);(1,3,4,7)(2,6,5,8)(9,12)(10,13)(11,19)(14,23)(15,20)(16,21)(17,24)(18,22)(25,29,31,33)(26,34,36,28)(27,35,38,40)(30,32,39,37);(9,13)(10,12)(11,20)(14,21)(15,19)(16,23)(17,22)(18,24)(25,30)(26,35)(27,28)(29,32)(31,39)(33,37)(34,38)(36,40)Polycyclic presentation:
2^9|a^2=h,b^2=gi,e^2=gh,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7673767);