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 | |
4487 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(4,7)(9,10,15,17)(11,16,12,22)(13,18,23,21)(14,20,24,19)(25,26,30,34)(27,35,37,31)(28,38,39,32)(29,33,36,40);(1,2,5,6)(3,4,8,7)(9,11)(10,16)(12,15)(13,20)(14,21)(17,22)(18,24)(19,23)(25,27,30,37)(26,31,34,35)(28,29,39,36)(32,33,38,40);(1,2,5,6)(3,4,8,7)(10,17)(11,12)(18,21)(19,20)(25,28,30,39)(26,32,34,38)(27,36,37,29)(31,40,35,33);(1,3)(2,7)(4,6)(5,8)(9,12,15,11)(10,16,17,22)(13,19,23,20)(14,18,24,21)(25,28)(26,32)(27,29)(30,39)(31,33)(34,38)(35,40)(36,37);(1,3)(2,4)(5,8)(6,7)(9,13)(10,18)(11,20)(12,19)(14,22)(15,23)(16,24)(17,21)(25,29)(26,33)(27,28)(30,36)(31,32)(34,40)(35,38)(37,39);(1,4)(2,8)(3,6)(5,7)(9,14,15,24)(10,19,17,20)(11,21,12,18)(13,16,23,22)(25,26)(27,31)(28,32)(29,33)(30,34)(35,37)(36,40)(38,39)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=g,d^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9845274);