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 | |
4693 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10,15,18)(11,17,12,16)(13,23,24,14)(19,22,21,20)(25,26,31,35)(27,32,29,30)(28,34,39,33)(36,38,37,40);(2,3)(6,7)(9,11,15,12)(10,16,18,17)(13,21,24,19)(14,20,23,22)(25,27)(26,32)(28,37)(29,31)(30,35)(33,38)(34,40)(36,39);(1,3,5,2)(4,7,8,6)(9,11,15,12)(10,17,18,16)(13,19,24,21)(14,20,23,22)(25,28)(26,33)(27,36)(29,37)(30,38)(31,39)(32,40)(34,35);(1,4,5,8)(2,6,3,7)(9,12,15,11)(10,17,18,16)(13,21,24,19)(14,20,23,22)(25,29,31,27)(26,30,35,32)(28,37,39,36)(33,38,34,40);(9,13)(10,14)(11,19)(12,21)(15,24)(16,22)(17,20)(18,23)(25,30)(26,27)(28,38)(29,35)(31,32)(33,36)(34,37)(39,40);(9,14)(10,13)(11,20)(12,22)(15,23)(16,21)(17,19)(18,24)(25,28)(26,34)(27,37)(29,36)(30,38)(31,39)(32,40)(33,35)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=g,d^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8383040);