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 | |
4440 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)(11,13)(12,20)(14,16)(15,18)(17,21)(19,23)(22,24);(1,3,5,8)(2,6,4,7)(9,11)(10,15)(12,19)(13,16)(14,18)(17,23)(20,22)(21,24)(25,26)(27,31)(28,35)(29,30)(32,36)(33,39)(34,37)(38,40);(1,2,5,4)(3,6,8,7)(10,16)(11,18)(12,21)(22,23)(26,30)(31,37)(32,35)(38,39);(1,2,5,4)(3,6,8,7)(9,11)(10,13)(12,19)(14,18)(15,16)(17,22)(20,23)(21,24)(25,27,29,34)(26,31,30,37)(28,33,36,40)(32,38,35,39);(1,4,5,2)(3,7,8,6)(9,12,14,21)(10,17,16,20)(11,19,18,24)(13,22,15,23)(25,26)(27,31)(28,32)(29,30)(33,38)(34,37)(35,36)(39,40);(9,13)(10,11)(12,22)(14,15)(16,18)(17,19)(20,24)(21,23)(25,28)(26,32)(27,33)(29,36)(30,35)(31,38)(34,40)(37,39)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=gh,d^2=gi,e^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7631762);