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 | |
4395 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(9,10)(11,17)(12,18)(13,21)(14,15)(16,22)(19,20)(23,24)(25,26)(27,32)(28,30)(29,38)(31,33)(34,37)(35,40)(36,39);(3,7)(6,8)(10,15)(11,18)(16,21)(19,24)(25,27,31,36)(26,32,33,39)(28,29,35,37)(30,38,40,34);(1,2,5,4)(3,6,7,8)(10,15)(11,18)(16,21)(19,24)(26,33)(27,36)(28,35)(34,38);(1,3)(2,6)(4,8)(5,7)(9,11)(10,12)(13,19)(14,18)(15,17)(16,20)(21,23)(22,24)(25,28,31,35)(26,30,33,40)(27,37,36,29)(32,34,39,38);(1,4,5,2)(3,8,7,6)(9,12)(10,11)(13,20)(14,17)(15,18)(16,19)(21,24)(22,23)(25,29,31,37)(26,34,33,38)(27,35,36,28)(30,39,40,32);(9,13,14,22)(10,16,15,21)(11,19,18,24)(12,20,17,23)(25,30)(26,35)(27,38)(28,33)(29,39)(31,40)(32,37)(34,36)Polycyclic presentation:
2^9|b^2=i,c^2=ghi,d^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8014822);