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 | #36157 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4534 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)(9,10,15,18)(11,17,14,16)(12,21,23,13)(19,20,22,24)(25,26,29,33)(27,31,34,30)(28,36,37,32)(35,38,40,39);(3,7)(5,8)(9,11,15,14)(10,16,18,17)(12,19,23,22)(13,20,21,24)(27,34)(30,31)(35,40)(38,39);(1,2)(3,5)(4,6)(7,8)(25,27)(26,30)(28,35)(29,34)(31,33)(32,38)(36,39)(37,40);(1,3)(2,5)(4,7)(6,8)(9,12,15,23)(10,13,18,21)(11,19,14,22)(16,20,17,24)(25,27)(26,31)(28,35)(29,34)(30,33)(32,39)(36,38)(37,40);(1,2)(3,5)(4,6)(7,8)(9,13)(10,12)(11,20)(14,24)(15,21)(16,19)(17,22)(18,23)(25,26)(27,30)(28,32)(29,33)(31,34)(35,38)(36,37)(39,40);(9,14,15,11)(10,17,18,16)(12,22,23,19)(13,24,21,20)(25,28,29,37)(26,32,33,36)(27,35,34,40)(30,38,31,39)Polycyclic presentation:
2^9|a^2=gh,b^2=g,d^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8055628);