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 | #51926 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4311 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)(3,4)(5,8)(6,7)(12,20)(15,22)(18,24)(21,23)(25,26,31,35)(27,34,30,32)(28,29,33,40)(36,37,38,39);(1,3)(2,4)(5,6)(7,8)(9,10)(11,16)(12,15)(13,17)(14,19)(18,23)(20,22)(21,24)(25,27,31,30)(26,32,35,34)(28,36,33,38)(29,39,40,37);(1,2,7,6)(3,5,8,4)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,27,31,30)(26,32,35,34)(28,38,33,36)(29,37,40,39);(1,4,7,5)(2,8,6,3)(9,12)(10,15)(11,18)(13,20)(14,21)(16,23)(17,22)(19,24)(25,28,31,33)(26,29,35,40)(27,36,30,38)(32,39,34,37);(1,5,7,4)(2,8,6,3)(9,11)(10,16)(12,18)(13,19)(14,17)(15,23)(20,24)(21,22)(25,29)(26,33)(27,37)(28,35)(30,39)(31,40)(32,36)(34,38);(1,6,7,2)(3,5,8,4)(9,10)(11,14)(12,15)(13,17)(16,19)(18,21)(20,22)(23,24)(25,30,31,27)(26,34,35,32)(28,38,33,36)(29,37,40,39)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=h,d^2=h,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9619990);