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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4720 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,19,20,16)(12,13,18,17)(21,22,23,24)(25,26,29,33)(27,31,35,30)(28,32,36,39)(34,38,40,37);(1,2)(3,7)(4,5)(6,8)(10,15)(11,20)(12,18)(22,24)(25,27,29,35)(26,30,33,31)(28,34,36,40)(32,37,39,38);(2,5)(6,7)(10,15)(13,17)(16,19)(22,24)(28,36)(32,39)(34,40)(37,38);(3,8)(6,7)(9,11)(10,16)(12,23)(13,22)(14,20)(15,19)(17,24)(18,21)(25,26)(27,30)(28,32)(29,33)(31,35)(34,37)(36,39)(38,40);(1,3)(2,6)(4,8)(5,7)(9,12,14,18)(10,17,15,13)(11,21,20,23)(16,22,19,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39);(1,2)(3,6)(4,5)(7,8)(9,13,14,17)(10,18,15,12)(11,22,20,24)(16,21,19,23)(25,28,29,36)(26,32,33,39)(27,34,35,40)(30,37,31,38)Polycyclic presentation:
2^9|a^2=g,b^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8254625);