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 | #2699 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4109 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,12)(13,17)(14,15)(16,19)(18,20)(21,24)(22,23)(25,26,32,36)(27,33,29,30)(28,37,39,34)(31,38,40,35);(2,5)(6,7)(10,15)(12,16)(17,23)(20,21)(25,27,32,29)(26,33,36,30)(28,38,39,35)(31,37,40,34);(1,2)(3,6)(4,5)(7,8)(25,28,32,39)(26,34,36,37)(27,35,29,38)(30,31,33,40);(9,11)(10,16)(12,15)(13,18)(14,19)(17,21)(20,23)(22,24)(25,29,32,27)(26,30,36,33)(28,35,39,38)(31,34,40,37);(1,2)(3,7)(4,5)(6,8)(9,12,14,16)(10,11,15,19)(13,21,22,20)(17,24,23,18)(25,30)(26,27)(28,31)(29,36)(32,33)(34,35)(37,38)(39,40);(1,3)(2,6)(4,8)(5,7)(9,13,14,22)(10,17,15,23)(11,18,19,24)(12,20,16,21)(25,31)(26,35)(27,37)(28,33)(29,34)(30,39)(32,40)(36,38)Polycyclic presentation:
2^9|a^2=h,b^2=h,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, 9686681);