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 | |
4487 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)(11,19)(12,17)(13,21)(14,15)(16,20)(18,23)(22,24)(25,26)(27,32)(28,29)(30,38)(31,33)(34,40)(35,37)(36,39);(1,2,5,8)(3,4,7,6)(10,15)(12,22)(16,19)(18,21)(25,27,31,28)(26,32,33,29)(30,35,40,36)(34,39,38,37);(1,3)(2,6)(4,8)(5,7)(10,15)(11,20)(17,24)(18,21)(26,33)(27,28)(34,38)(35,36);(1,4)(2,7)(3,8)(5,6)(9,11)(10,16)(12,18)(13,17)(14,20)(15,19)(21,22)(23,24)(25,28,31,27)(26,29,33,32)(30,36,40,35)(34,37,38,39);(1,3)(2,4)(5,7)(6,8)(9,12,14,22)(10,17,15,24)(11,21,20,18)(13,16,23,19)(25,29)(26,28)(27,33)(30,39)(31,32)(34,35)(36,38)(37,40);(1,4)(2,7)(3,8)(5,6)(9,13,14,23)(10,18,15,21)(11,17,20,24)(12,19,22,16)(25,30,31,40)(26,34,33,38)(27,35,28,36)(29,37,32,39)Polycyclic presentation:
2^9|b^2=i,d^2=gh,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10004133);