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:
(1,2,5,4)(3,7,8,6)(9,10)(11,12)(13,19)(14,16)(15,17)(18,20)(21,24)(22,23)(25,26)(27,34)(28,31)(29,36)(30,33)(32,38)(35,37)(39,40);(1,3)(2,6)(4,7)(5,8)(9,11)(10,13)(12,17)(14,18)(15,19)(16,21)(20,23)(22,24)(26,28)(27,33)(32,36)(35,39);(2,4)(6,7)(9,12,15,13)(10,11,17,19)(14,20,22,21)(16,18,23,24)(25,27)(26,30)(28,34)(29,35)(31,33)(32,37)(36,40)(38,39);(1,3)(2,6)(4,7)(5,8)(9,13,15,12)(10,11,17,19)(14,21,22,20)(16,18,23,24)(25,28,31,26)(27,34,33,30)(29,36,38,32)(35,40,39,37);(1,3)(2,6)(4,7)(5,8)(9,14,15,22)(10,16,17,23)(11,18,19,24)(12,20,13,21)(25,29)(26,32)(27,35)(28,36)(30,37)(31,38)(33,39)(34,40);(1,4,5,2)(3,7,8,6)(9,11)(10,12)(13,17)(14,18)(15,19)(16,20)(21,23)(22,24)(25,30,31,34)(26,33,28,27)(29,37,38,40)(32,39,36,35)Polycyclic presentation:
2^9|a^2=hi,c^2=gh,d^2=ghi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8095209);