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 | |
4531 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)(12,20)(15,22)(18,24)(21,23)(25,26)(27,33)(28,35)(29,37)(30,32)(31,38)(34,36)(39,40);(1,3)(2,4)(5,8)(6,7)(9,10)(11,14)(12,15)(13,16)(17,19)(18,21)(20,22)(23,24)(25,27)(26,28)(29,34)(30,35)(31,36)(32,33)(37,39)(38,40);(1,4,6,5)(2,7,8,3)(9,10,13,16)(11,14,19,17)(12,15,20,22)(18,21,24,23)(25,26,30,32)(27,28,35,33)(29,31,38,37)(34,36,40,39);(2,8)(4,5)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,28,30,33)(26,27,32,35)(29,36,38,39)(31,34,37,40);(1,5,6,4)(2,3,8,7)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,27)(26,28)(29,34)(30,35)(31,36)(32,33)(37,39)(38,40);(1,3)(2,4)(5,8)(6,7)(9,12)(10,15)(11,18)(13,20)(14,21)(16,22)(17,23)(19,24)(25,29)(26,31)(27,34)(28,36)(30,38)(32,37)(33,39)(35,40)Polycyclic presentation:
2^9|c^2=gi,d^2=h,e^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, 8064443);