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 | |
4639 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)(3,7)(4,5)(6,8)(9,10)(11,15)(12,17)(13,21)(14,16)(18,23)(19,22)(20,24)(25,26)(27,30)(28,35)(29,32)(31,36)(33,37)(34,40)(38,39);(2,5)(6,7)(9,11,14,19)(10,15,16,22)(12,18,20,21)(13,17,23,24)(27,33)(30,37)(34,39)(38,40);(1,2)(3,6)(4,5)(7,8)(10,16)(11,19)(12,20)(13,23)(25,26,29,32)(27,30,33,37)(28,31,36,35)(34,38,39,40);(1,3,4,8)(2,6,5,7)(9,12)(10,17)(11,18)(13,22)(14,20)(15,23)(16,24)(19,21)(25,26)(27,30)(28,31)(29,32)(33,37)(34,38)(35,36)(39,40);(9,11)(10,15)(12,21)(13,17)(14,19)(16,22)(18,20)(23,24)(25,27)(26,30)(28,34)(29,33)(31,38)(32,37)(35,40)(36,39);(9,13)(10,18)(11,17)(12,22)(14,23)(15,20)(16,21)(19,24)(25,28)(26,31)(27,34)(29,36)(30,38)(32,35)(33,39)(37,40)Polycyclic presentation:
2^9|b^2=gi,c^2=ghi,d^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9401828);