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 | #37839 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4492 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)(3,5)(4,7)(6,8)(10,13)(11,17)(16,21)(18,20)(25,26,31,29)(27,32,35,37)(28,33,38,39)(30,40,36,34);(2,3)(6,7)(9,10,14,13)(11,15,17,19)(12,16,22,21)(18,23,20,24)(25,27,31,35)(26,32,29,37)(28,36,38,30)(33,34,39,40);(1,2,5,3)(4,6,8,7)(11,17)(15,19)(18,20)(23,24)(27,35)(28,38)(32,37)(33,39);(1,3,5,2)(4,7,8,6)(9,11,14,17)(10,15,13,19)(12,20,22,18)(16,24,21,23)(25,28)(26,33)(27,36)(29,39)(30,35)(31,38)(32,34)(37,40);(1,4)(2,6)(3,7)(5,8)(9,12,14,22)(10,16,13,21)(11,18,17,20)(15,23,19,24)(25,29,31,26)(27,37,35,32)(28,33,38,39)(30,40,36,34);(9,13,14,10)(11,19,17,15)(12,21,22,16)(18,24,20,23)(25,30,31,36)(26,34,29,40)(27,28,35,38)(32,39,37,33)Polycyclic presentation:
2^9|a^2=hi,b^2=gi,c^2=h,d^2=g,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8082933);