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 | #50005 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4354 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10)(11,14)(12,20)(13,15)(16,18)(17,21)(19,23)(22,24)(25,26,31,36)(27,37,39,32)(28,35,30,33)(29,34,38,40);(3,7)(6,8)(9,11)(10,14)(12,19)(13,18)(15,16)(17,22)(20,23)(21,24)(25,26)(27,37)(28,33)(29,40)(30,35)(31,36)(32,39)(34,38);(1,2)(3,6)(4,5)(7,8)(10,15)(11,18)(12,21)(22,23)(25,27,31,39)(26,32,36,37)(28,38,30,29)(33,40,35,34);(9,11)(10,16)(12,19)(13,18)(14,15)(17,23)(20,22)(21,24)(25,28,31,30)(26,33,36,35)(27,38,39,29)(32,40,37,34);(1,3)(2,6)(4,7)(5,8)(9,10)(11,14)(12,17)(13,15)(16,18)(19,22)(20,21)(23,24)(25,29)(26,34)(27,28)(30,39)(31,38)(32,33)(35,37)(36,40);(1,2)(3,6)(4,5)(7,8)(9,12)(10,17)(11,19)(13,21)(14,22)(15,20)(16,23)(18,24)(25,30,31,28)(26,35,36,33)(27,29,39,38)(32,34,37,40)Polycyclic presentation:
2^9|a^2=g,c^2=g,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8948262);