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 | #34589 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4345 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,13)(12,20)(14,15)(16,18)(17,21)(19,24)(22,23)(25,26)(27,34)(28,31)(29,32)(30,33)(35,37)(36,39)(38,40);(1,3,4,8)(2,5,6,7)(10,15)(11,18)(12,21)(23,24)(25,27)(26,28)(29,35)(30,31)(32,36)(33,34)(37,40)(38,39);(1,3)(2,5)(4,8)(6,7)(10,15)(13,16)(17,20)(23,24)(25,28,30,34)(26,31,33,27)(29,37,38,36)(32,35,40,39);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,22)(13,15)(14,18)(17,23)(19,21)(20,24)(25,27)(26,28)(29,35)(30,31)(32,36)(33,34)(37,40)(38,39);(1,2)(3,5)(4,6)(7,8)(9,12)(10,17)(11,19)(13,23)(14,21)(15,20)(16,24)(18,22)(25,26)(27,34)(28,31)(29,32)(30,33)(35,37)(36,39)(38,40);(9,13,14,16)(10,11,15,18)(12,23,21,24)(17,19,20,22)(25,29,30,38)(26,32,33,40)(27,35,31,39)(28,36,34,37)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,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, 8282828);