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 | #46812 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4183 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,16)(12,20)(13,17)(14,18)(15,22)(19,21)(23,24)(25,26)(27,35)(28,31)(29,33)(30,38)(32,36)(34,39)(37,40);(1,2)(3,8)(4,6)(5,7)(9,11)(10,13)(12,19)(14,17)(15,21)(16,18)(20,23)(22,24)(25,27,31,36)(26,32,28,35)(29,39,40,30)(33,38,37,34);(1,3,5,6)(2,4,7,8)(26,28)(32,35)(33,37)(34,38);(9,12,14,22)(10,15,18,20)(11,19,17,24)(13,21,16,23)(25,28,31,26)(27,35,36,32)(29,37,40,33)(30,38,39,34);(1,2)(3,4)(5,7)(6,8)(9,11)(10,16)(12,19)(13,18)(14,17)(15,23)(20,21)(22,24)(25,29,31,40)(26,33,28,37)(27,30,36,39)(32,34,35,38);(1,4)(2,6)(3,7)(5,8)(9,13,14,16)(10,17,18,11)(12,21,22,23)(15,24,20,19)(25,30)(26,34)(27,29)(28,38)(31,39)(32,33)(35,37)(36,40)Polycyclic presentation:
2^9|b^2=h,c^2=i,d^2=g,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,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, 9591153);