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 | |
4705 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)(4,8)(6,7)(9,10)(11,18)(12,15)(13,21)(14,17)(16,22)(19,23)(20,24)(25,26,30,29)(27,33,35,31)(28,36,37,32)(34,38,40,39);(2,7)(3,4)(11,12)(13,22)(15,18)(16,21)(25,27)(26,31)(28,34)(29,33)(30,35)(32,38)(36,39)(37,40);(2,7)(5,8)(9,11,14,12)(10,15,17,18)(13,19,22,20)(16,23,21,24)(25,26,30,29)(27,31,35,33)(28,32,37,36)(34,38,40,39);(1,3,6,4)(2,8,7,5)(9,12,14,11)(10,15,17,18)(13,20,22,19)(16,23,21,24)(25,28)(26,32)(27,34)(29,36)(30,37)(31,38)(33,39)(35,40);(1,4,6,3)(2,5,7,8)(9,13)(10,16)(11,19)(12,20)(14,22)(15,23)(17,21)(18,24)(25,29,30,26)(27,31,35,33)(28,36,37,32)(34,38,40,39);(1,5)(2,3)(4,7)(6,8)(9,10)(11,18)(12,15)(13,16)(14,17)(19,24)(20,23)(21,22)(25,27)(26,33)(28,34)(29,31)(30,35)(32,39)(36,38)(37,40)Polycyclic presentation:
2^9|a^2=ghi,c^2=gh,d^2=h,e^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=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8109574);