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 | |
4744 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)(5,8)(10,15)(11,18)(17,23)(19,21)(25,26)(27,32)(28,29)(30,34)(31,35)(33,40)(36,39)(37,38);(1,2,4,6)(3,5,7,8)(9,10)(11,16)(12,17)(13,18)(14,15)(19,24)(20,23)(21,22)(27,35)(31,32)(36,38)(37,39);(1,3)(2,5)(4,7)(6,8)(11,18)(12,20)(13,16)(17,23)(25,27,30,35)(26,31,34,32)(28,38,33,36)(29,39,40,37);(9,11,14,18)(10,16,15,13)(12,21,20,19)(17,22,23,24)(25,28,30,33)(26,29,34,40)(27,36,35,38)(31,37,32,39);(1,2)(3,5)(4,6)(7,8)(9,12,14,20)(10,17,15,23)(11,19,18,21)(13,22,16,24)(25,27)(26,32)(28,38)(29,37)(30,35)(31,34)(33,36)(39,40);(1,3)(2,5)(4,7)(6,8)(9,13,14,16)(10,18,15,11)(12,22,20,24)(17,21,23,19)(25,29)(26,33)(27,37)(28,34)(30,40)(31,38)(32,36)(35,39)Polycyclic presentation:
2^9|b^2=hi,c^2=i,d^2=gi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8901252);