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 | #2410 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4404 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,15)(12,17)(13,23)(14,16)(18,20)(19,24)(21,22)(25,26,31,35)(27,36,37,32)(28,30,34,40)(29,38,39,33);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,20)(13,21)(14,19)(16,24)(17,18)(22,23)(25,27)(26,32)(28,38)(29,30)(31,37)(33,34)(35,36)(39,40);(1,3,4,7)(2,5,6,8)(10,16)(11,19)(13,20)(17,22)(27,37)(29,39)(32,36)(33,38);(9,12,14,21)(10,17,16,22)(11,20,19,13)(15,18,24,23)(25,28)(26,30)(27,38)(29,32)(31,34)(33,37)(35,40)(36,39);(9,13,14,20)(10,18,16,23)(11,21,19,12)(15,17,24,22)(25,29)(26,33)(27,30)(28,36)(31,39)(32,34)(35,38)(37,40);(9,10)(11,15)(12,22)(13,18)(14,16)(17,21)(19,24)(20,23)(25,30)(26,34)(27,29)(28,35)(31,40)(32,33)(36,38)(37,39)Polycyclic presentation:
2^9|a^2=g,c^2=ghi,d^2=h,e^2=h,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7982615);