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 |
| prime radical | #37722 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4683 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,18)(12,22)(13,17)(14,15)(16,21)(19,24)(20,23)(25,26)(27,28)(29,32)(30,38)(31,34)(33,40)(35,37)(36,39);(1,2,5,6)(3,7,8,4)(10,15)(12,17)(16,18)(19,23)(25,27,31,32)(26,28,34,29)(30,39,40,35)(33,37,38,36);(2,6)(4,7)(10,15)(13,22)(16,18)(20,24)(25,28,31,29)(26,32,34,27)(30,36,40,37)(33,39,38,35);(2,6)(3,8)(9,11,14,21)(10,16,15,18)(12,23,17,19)(13,20,22,24)(25,29,31,28)(26,32,34,27)(30,36,40,37)(33,35,38,39);(1,3)(2,4)(5,8)(6,7)(9,12,14,17)(10,13,15,22)(11,19,21,23)(16,24,18,20)(25,30)(26,33)(27,35)(28,36)(29,37)(31,40)(32,39)(34,38);(1,4)(2,3)(5,7)(6,8)(9,13)(10,17)(11,20)(12,15)(14,22)(16,19)(18,23)(21,24)(25,26)(27,28)(29,32)(30,33)(31,34)(35,36)(37,39)(38,40)Polycyclic presentation:
2^9|b^2=i,c^2=gh,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8295160);