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 | |
4572 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,12)(13,16)(14,17)(15,18)(19,20)(21,22)(23,24)(25,26)(27,34)(28,37)(29,32)(30,36)(31,33)(35,39)(38,40);(1,2,4,6)(3,5,8,7)(9,11)(10,12)(13,19)(14,15)(16,20)(17,18)(21,24)(22,23)(25,27)(26,30)(28,38)(29,35)(31,36)(32,40)(33,34)(37,39);(1,2)(3,5)(4,6)(7,8)(9,10)(11,18)(12,15)(13,16)(14,17)(19,24)(20,23)(21,22)(25,28,31,29)(26,32,33,37)(27,35,36,38)(30,39,34,40);(3,8)(5,7)(9,10)(11,12)(13,21)(14,17)(15,18)(16,22)(19,24)(20,23)(25,27)(26,30)(28,35)(29,38)(31,36)(32,39)(33,34)(37,40);(1,2)(3,5)(4,6)(7,8)(9,12)(10,15)(11,17)(13,20)(14,18)(16,23)(19,21)(22,24)(25,29,31,28)(26,32,33,37)(27,35,36,38)(30,40,34,39);(1,3)(2,5)(4,8)(6,7)(9,13)(10,16)(11,19)(12,20)(14,22)(15,23)(17,21)(18,24)(25,30)(26,27)(28,39)(29,40)(31,34)(32,35)(33,36)(37,38)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10315274);