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 | #54285 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4204 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,5)(4,7)(6,8)(9,10,15,13)(11,18,14,16)(12,20,21,17)(19,23,22,24)(25,26,31,27)(28,35,38,32)(29,33,30,34)(36,40,37,39);(1,2,5,3)(4,6,8,7)(9,11,15,14)(10,16,13,18)(12,19,21,22)(17,23,20,24)(26,27)(32,35)(33,34)(39,40);(1,2,5,3)(4,6,8,7)(12,21)(17,20)(19,22)(23,24)(25,27,31,26)(28,35,38,32)(29,34,30,33)(36,40,37,39);(2,3)(6,7)(9,12)(10,17)(11,19)(13,20)(14,22)(15,21)(16,23)(18,24)(25,28)(26,32)(27,35)(29,36)(30,37)(31,38)(33,39)(34,40);(1,3,5,2)(4,7,8,6)(9,13,15,10)(11,18,14,16)(12,17,21,20)(19,23,22,24)(25,29,31,30)(26,33,27,34)(28,36,38,37)(32,39,35,40);(1,4)(2,6)(3,7)(5,8)(9,14,15,11)(10,18,13,16)(12,22,21,19)(17,24,20,23)(25,30,31,29)(26,34,27,33)(28,37,38,36)(32,40,35,39)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=hi,e^2=g,f^2=gh,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, 8066164);