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 | |
4430 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)(6,8)(9,10)(11,14)(12,19)(13,16)(15,20)(17,21)(18,24)(22,23)(25,26)(27,31)(28,36)(29,33)(30,35)(32,37)(34,38)(39,40);(1,2,4,5)(3,6,7,8)(11,17)(14,21)(18,23)(22,24)(25,27,29,35)(26,30,33,31)(28,34,37,40)(32,38,36,39);(1,2)(3,6)(4,5)(7,8)(9,10,13,16)(11,14,17,21)(12,15,20,19)(18,22,23,24)(25,27,29,35)(26,31,33,30)(28,34,37,40)(32,39,36,38);(2,5)(6,8)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,20)(23,24)(25,27)(26,30)(28,34)(29,35)(31,33)(32,38)(36,39)(37,40);(1,2)(3,8)(4,5)(6,7)(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,26)(27,31)(28,36)(29,33)(30,35)(32,37)(34,38)(39,40);(1,3)(2,6)(4,7)(5,8)(9,12)(10,15)(11,18)(13,20)(14,22)(16,19)(17,23)(21,24)(25,28,29,37)(26,32,33,36)(27,34,35,40)(30,38,31,39)Polycyclic presentation:
2^9|b^2=gi,c^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10219767);