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 | |
4674 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(5,8)(9,10)(11,12)(13,16)(14,15)(17,19)(18,20)(21,22)(23,24)(25,26)(27,31)(28,35)(29,37)(30,33)(32,39)(34,36)(38,40);(1,2,4,6)(3,5,7,8)(10,15)(12,19)(16,22)(20,24)(25,27,30,35)(26,31,33,28)(29,34,39,40)(32,38,37,36);(1,3)(2,5)(4,7)(6,8)(11,17)(12,19)(13,21)(16,22)(25,28,30,31)(26,27,33,35)(29,38,39,36)(32,40,37,34);(9,11,14,17)(10,12,15,19)(13,18,21,23)(16,20,22,24)(25,28)(26,27)(29,36)(30,31)(32,34)(33,35)(37,40)(38,39);(1,2)(3,5)(4,6)(7,8)(9,12)(10,11)(13,20)(14,19)(15,17)(16,18)(21,24)(22,23)(25,26)(27,31)(28,35)(29,32)(30,33)(34,38)(36,40)(37,39);(1,3)(2,5)(4,7)(6,8)(9,13,14,21)(10,16,15,22)(11,18,17,23)(12,20,19,24)(25,29,30,39)(26,32,33,37)(27,34,35,40)(28,36,31,38)Polycyclic presentation:
2^9|b^2=h,c^2=i,d^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8964966);