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 | |
4662 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10)(11,14)(12,20)(13,16)(15,21)(17,19)(18,23)(22,24)(25,26,30,33)(27,34,29,31)(28,36,38,32)(35,39,37,40);(1,3,5,8)(2,6,7,4)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,21)(22,24)(25,26)(27,31)(28,32)(29,34)(30,33)(35,39)(36,38)(37,40);(1,2)(3,4)(5,7)(6,8)(9,10,13,16)(11,14,19,17)(12,20,21,15)(18,23,24,22)(25,27,30,29)(26,31,33,34)(28,37,38,35)(32,40,36,39);(1,4)(2,3)(5,6)(7,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,21)(22,24)(25,28,30,38)(26,32,33,36)(27,35,29,37)(31,39,34,40);(1,3)(2,4)(5,8)(6,7)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,29,30,27)(26,31,33,34)(28,35,38,37)(32,40,36,39);(1,4)(2,3)(5,6)(7,8)(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(25,26)(27,34)(28,32)(29,31)(30,33)(35,40)(36,38)(37,39)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=gi,d^2=hi,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8831967);