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 | #47939 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4398 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)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,26)(27,29)(28,36)(30,32)(31,35)(33,37)(34,39)(38,40);(1,2,5,3)(4,7,8,6)(11,17)(15,21)(18,23)(22,24)(25,27,30,35)(26,31,32,29)(28,34,37,40)(33,39,36,38);(2,3)(6,7)(10,14)(15,21)(16,19)(22,24)(26,32)(28,37)(29,31)(34,40);(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,28)(26,33)(27,34)(29,38)(30,37)(31,39)(32,36)(35,40);(1,3,5,2)(4,7,8,6)(9,11,13,17)(10,15,14,21)(12,18,20,23)(16,22,19,24)(25,29,30,31)(26,27,32,35)(28,38,37,39)(33,34,36,40);(1,4,5,8)(2,6,3,7)(9,12,13,20)(10,16,14,19)(11,18,17,23)(15,22,21,24)(25,27)(26,31)(28,34)(29,32)(30,35)(33,39)(36,38)(37,40)Polycyclic presentation:
2^9|b^2=gi,e^2=hi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9253884);