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 | |
4799 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,7)(9,10)(11,14)(12,15)(13,18)(16,21)(17,20)(19,23)(22,24)(25,26)(27,35)(28,30)(29,32)(31,33)(34,40)(36,39)(37,38);(1,2,5,4)(3,7,8,6)(9,10,14,11)(12,17,20,15)(13,16,21,18)(19,23,24,22)(25,27,31,29)(26,32,33,35)(28,36,34,38)(30,37,40,39);(1,3,5,8)(2,6,4,7)(10,11)(15,17)(16,18)(22,23)(26,33)(28,34)(32,35)(36,38);(1,4,5,2)(3,7,8,6)(9,11,14,10)(12,17,20,15)(13,18,21,16)(19,23,24,22)(25,28,31,34)(26,30,33,40)(27,36,29,38)(32,37,35,39);(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23)(25,29,31,27)(26,32,33,35)(28,38,34,36)(30,37,40,39);(9,13)(10,16)(11,18)(12,19)(14,21)(15,22)(17,23)(20,24)(25,30)(26,34)(27,37)(28,33)(29,39)(31,40)(32,38)(35,36)Polycyclic presentation:
2^9|b^2=g,c^2=gh,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9818107);