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 | #52394 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4464 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)(11,18)(12,20)(13,17)(14,15)(16,23)(19,22)(21,24)(25,26,30,33)(27,28,35,31)(29,32,37,39)(34,36,40,38);(1,2)(3,6)(4,5)(7,8)(10,15)(11,17)(16,20)(19,21)(25,27)(26,31)(28,33)(29,34)(30,35)(32,38)(36,39)(37,40);(2,5)(6,8)(10,15)(13,18)(16,20)(22,24)(29,37)(32,39)(34,40)(36,38);(3,7)(6,8)(9,11,14,17)(10,13,15,18)(12,21,23,19)(16,24,20,22)(25,28,30,31)(26,27,33,35)(29,36,37,38)(32,34,39,40);(1,3)(2,6)(4,7)(5,8)(9,12,14,23)(10,16,15,20)(11,19,17,21)(13,22,18,24)(25,27)(26,28)(29,34)(30,35)(31,33)(32,36)(37,40)(38,39);(1,2)(3,6)(4,5)(7,8)(9,13)(10,17)(11,15)(12,22)(14,18)(16,21)(19,20)(23,24)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40)Polycyclic presentation:
2^9|a^2=gh,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8180594);