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 | #2682 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4100 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,15)(11,18)(17,20)(19,24)(25,26,31,35)(27,32,37,29)(28,38,36,33)(30,40,39,34);(2,5)(6,7)(9,10,14,15)(11,16,18,13)(12,20,22,17)(19,23,24,21)(25,27)(26,32)(28,39)(29,35)(30,36)(31,37)(33,40)(34,38);(1,2)(3,6)(4,5)(7,8)(25,28,31,36)(26,33,35,38)(27,30,37,39)(29,40,32,34);(9,11)(10,16)(12,19)(13,15)(14,18)(17,21)(20,23)(22,24)(25,29,31,32)(26,27,35,37)(28,34,36,40)(30,33,39,38);(1,2)(3,7)(4,5)(6,8)(9,12,14,22)(10,17,15,20)(11,19,18,24)(13,23,16,21)(25,27)(26,32)(28,30)(29,35)(31,37)(33,34)(36,39)(38,40);(1,3)(2,6)(4,8)(5,7)(9,13,14,16)(10,11,15,18)(12,21,22,23)(17,24,20,19)(25,30)(26,34)(27,36)(28,37)(29,33)(31,39)(32,38)(35,40)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=h,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,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, 9686556);