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 | #40417 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4249 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10,14,15)(11,17,19,12)(13,21,22,16)(18,20,24,23)(26,31)(29,32)(33,38)(37,39);(1,3,5,8)(2,6,7,4)(10,15)(12,17)(13,22)(18,24)(28,36)(33,38)(35,40)(37,39);(1,2)(3,4)(5,7)(6,8)(9,11,14,19)(10,12,15,17)(13,18,22,24)(16,20,21,23)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,38)(37,40);(1,4)(2,3)(5,6)(7,8)(9,11)(10,12)(13,18)(14,19)(15,17)(16,20)(21,23)(22,24)(25,27,30,34)(26,32,31,29)(28,35,36,40)(33,39,38,37);(1,3)(2,4)(5,8)(6,7)(9,12)(10,11)(13,20)(14,17)(15,19)(16,18)(21,24)(22,23)(25,28,30,36)(26,33,31,38)(27,35,34,40)(29,37,32,39);(1,4)(2,3)(5,6)(7,8)(9,13,14,22)(10,16,15,21)(11,18,19,24)(12,20,17,23)(25,29,30,32)(26,34,31,27)(28,37,36,39)(33,40,38,35)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=gi,d^2=gh,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8721159);