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 |
| prime radical | #40783 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4579 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,19)(12,21)(13,22)(14,18)(15,16)(17,20)(23,24)(25,26,30,35)(27,29,34,31)(28,32,38,33)(36,37,40,39);(1,2,5,8)(3,4,7,6)(9,11,15,21)(10,12,16,19)(13,14,20,23)(17,18,22,24)(25,27)(26,31)(28,36)(29,35)(30,34)(32,39)(33,37)(38,40);(1,3)(2,6)(4,8)(5,7)(10,16)(12,19)(17,22)(18,24)(25,28,30,38)(26,32,35,33)(27,36,34,40)(29,37,31,39);(1,4,5,6)(2,7,8,3)(9,12)(10,11)(13,18)(14,17)(15,19)(16,21)(20,24)(22,23)(25,28)(26,33)(27,36)(29,39)(30,38)(31,37)(32,35)(34,40);(1,3)(2,4)(5,7)(6,8)(9,13)(10,17)(11,14)(12,18)(15,20)(16,22)(19,24)(21,23)(25,27)(26,29)(28,36)(30,34)(31,35)(32,37)(33,39)(38,40);(1,4)(2,7)(3,8)(5,6)(9,14)(10,18)(11,20)(12,22)(13,21)(15,23)(16,24)(17,19)(25,29)(26,34)(27,35)(28,37)(30,31)(32,40)(33,36)(38,39)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=gh,d^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8623945);