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 | #2617 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4402 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)(3,7,8,5)(9,10,15,17)(11,18,22,16)(12,14,13,23)(19,24,20,21)(29,37)(32,39)(35,38)(36,40);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,19)(13,20)(14,21)(15,22)(17,18)(23,24)(25,26)(27,34)(28,31)(29,32)(30,33)(35,40)(36,38)(37,39);(1,3)(2,5)(4,8)(6,7)(9,12,15,13)(10,14,17,23)(11,19,22,20)(16,21,18,24)(25,27,30,31)(26,28,33,34)(29,35,37,38)(32,36,39,40);(1,3)(2,5)(4,8)(6,7)(9,13,15,12)(10,14,17,23)(11,20,22,19)(16,21,18,24)(25,26)(27,28)(29,32)(30,33)(31,34)(35,36)(37,39)(38,40);(1,2)(3,5)(4,6)(7,8)(9,14)(10,13)(11,21)(12,17)(15,23)(16,20)(18,19)(22,24)(25,28)(26,31)(27,33)(29,36)(30,34)(32,38)(35,39)(37,40);(9,11)(10,16)(12,19)(13,20)(14,21)(15,22)(17,18)(23,24)(25,29)(26,32)(27,35)(28,36)(30,37)(31,38)(33,39)(34,40)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=gi,d^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8044758);