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 | |
4448 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)(3,5,7,8)(9,10)(11,16)(12,20)(13,18)(14,15)(17,22)(19,21)(23,24)(25,26,31,32)(27,29,28,35)(30,33,39,40)(34,37,36,38);(3,7)(5,8)(10,15)(11,19)(17,20)(18,24)(25,27)(26,29)(28,31)(30,34)(32,35)(33,37)(36,39)(38,40);(1,2)(3,5)(4,6)(7,8)(10,15)(11,19)(17,20)(18,24)(26,32)(27,28)(33,40)(34,36);(1,3,4,7)(2,5,6,8)(9,11,14,19)(10,16,15,21)(12,18,22,24)(13,17,23,20)(25,28,31,27)(26,29,32,35)(30,34,39,36)(33,38,40,37);(1,2)(3,5)(4,6)(7,8)(9,12,14,22)(10,17,15,20)(11,18,19,24)(13,16,23,21)(25,29)(26,27)(28,32)(30,38)(31,35)(33,36)(34,40)(37,39);(9,13,14,23)(10,18,15,24)(11,20,19,17)(12,21,22,16)(25,30,31,39)(26,33,32,40)(27,34,28,36)(29,37,35,38)Polycyclic presentation:
2^9|a^2=gh,d^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8653557);