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 | #48109 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4466 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)(3,4)(5,6)(7,8)(9,10,14,11)(12,15,19,17)(13,18,21,16)(20,23,24,22)(25,26)(27,36)(28,29)(30,40)(31,35)(32,38)(33,39)(34,37);(1,3)(2,4)(5,8)(6,7)(12,19)(15,17)(20,24)(22,23)(25,27)(26,32)(28,30)(29,34)(31,38)(33,40)(35,36)(37,39);(1,4)(2,7)(3,5)(6,8)(10,11)(15,17)(16,18)(22,23)(25,28,31,39)(26,33,35,29)(27,30,38,37)(32,40,36,34);(1,2,6,5)(3,8,7,4)(9,11,14,10)(12,17,19,15)(13,18,21,16)(20,23,24,22)(25,28)(26,29)(27,37)(30,38)(31,39)(32,40)(33,35)(34,36);(1,4)(2,3)(5,7)(6,8)(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24)(25,29)(26,28)(27,34)(30,32)(31,33)(35,39)(36,37)(38,40);(1,5,6,2)(3,8,7,4)(9,13,14,21)(10,16,11,18)(12,20,19,24)(15,22,17,23)(25,30,31,37)(26,34,35,40)(27,28,38,39)(29,36,33,32)Polycyclic presentation:
2^9|a^2=ghi,c^2=g,d^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,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, 9845300);