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 | #59019 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4279 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,7,4)(9,10)(11,17)(12,16)(13,15)(14,19)(18,24)(20,22)(21,23)(28,36)(31,38)(35,40)(37,39);(1,3)(2,6)(4,8)(5,7)(9,11)(10,14)(12,20)(13,19)(15,17)(16,24)(18,21)(22,23)(25,26)(27,32)(28,31)(29,33)(30,34)(35,39)(36,38)(37,40);(1,2)(3,4)(5,8)(6,7)(10,15)(14,17)(16,23)(22,24)(25,27)(26,29)(28,35)(30,33)(31,37)(32,34)(36,40)(38,39);(1,4)(2,7)(3,8)(5,6)(9,10)(11,14)(12,16)(13,15)(17,19)(18,22)(20,24)(21,23)(25,28)(26,31)(27,35)(29,37)(30,36)(32,39)(33,40)(34,38);(1,4)(2,3)(5,6)(7,8)(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(25,27)(26,32)(28,35)(29,34)(30,33)(31,39)(36,40)(37,38);(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,20)(13,19)(15,17)(16,24)(18,21)(22,23)(25,29,30,32)(26,33,34,27)(28,37,36,39)(31,40,38,35)Polycyclic presentation:
2^9|a^2=gh,e^2=g,f^2=ghi,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, 9622653);