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 | #40943 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4324 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)(3,5,7,8)(11,17)(14,18)(19,23)(22,24)(25,26)(27,33)(28,36)(29,34)(30,31)(32,38)(35,37)(39,40);(1,3,4,7)(2,5,6,8)(9,10)(11,14)(12,20)(13,16)(15,21)(17,18)(19,24)(22,23)(26,31)(27,34)(32,36)(35,40);(1,2)(3,5)(4,6)(7,8)(9,10,13,16)(11,14,17,18)(12,15,21,20)(19,22,23,24)(25,27)(26,29)(28,35)(30,34)(31,33)(32,37)(36,39)(38,40);(9,10)(11,18)(12,15)(13,16)(14,17)(19,24)(20,21)(22,23)(25,28,30,38)(26,32,31,36)(27,35,34,40)(29,37,33,39);(1,3)(2,5)(4,7)(6,8)(9,11)(10,14)(12,19)(13,17)(15,22)(16,18)(20,24)(21,23)(25,26)(27,29)(28,32)(30,31)(33,34)(35,37)(36,38)(39,40);(1,2)(3,5)(4,6)(7,8)(9,12)(10,15)(11,19)(13,21)(14,22)(16,20)(17,23)(18,24)(25,29)(26,27)(28,37)(30,33)(31,34)(32,35)(36,40)(38,39)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=gi,d^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8736284);