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 | #37751 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4431 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)(17,24)(18,22)(21,23)(25,26)(27,33)(28,36)(29,32)(30,35)(31,38)(34,39)(37,40);(1,2)(3,7)(4,5)(6,8)(10,14)(12,20)(15,22)(17,21)(25,27,29,35)(26,30,32,33)(28,37,38,34)(31,40,36,39);(2,5)(6,7)(9,11,13,18)(10,15,14,22)(12,17,20,21)(16,23,19,24)(25,26,29,32)(27,30,35,33)(28,31,38,36)(34,39,37,40);(2,5)(6,7)(9,12,13,20)(10,16,14,19)(11,17,18,21)(15,23,22,24)(25,27)(26,30)(28,34)(29,35)(31,39)(32,33)(36,40)(37,38);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,21)(13,18)(14,22)(16,24)(17,20)(19,23)(25,26)(27,30)(28,31)(29,32)(33,35)(34,39)(36,38)(37,40);(1,3)(2,6)(4,8)(5,7)(9,10)(11,15)(12,16)(13,14)(17,23)(18,22)(19,20)(21,24)(25,28,29,38)(26,31,32,36)(27,34,35,37)(30,39,33,40)Polycyclic presentation:
2^9|b^2=h,c^2=g,d^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10230296);