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 | #51940 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4594 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10,14,15)(11,17,18,12)(13,21,22,16)(19,20,24,23)(25,26)(27,32)(28,34)(29,30)(31,33)(35,39)(36,38)(37,40);(3,7)(6,8)(10,15)(12,17)(16,21)(20,23)(25,27,31,36)(26,32,33,38)(28,35,37,29)(30,34,39,40);(1,2)(3,6)(4,5)(7,8)(10,15)(11,18)(13,22)(20,23)(26,33)(27,36)(28,37)(30,39);(9,11,14,18)(10,12,15,17)(13,19,22,24)(16,20,21,23)(25,28)(26,34)(27,35)(29,36)(30,38)(31,37)(32,39)(33,40);(1,3)(2,6)(4,7)(5,8)(9,12)(10,11)(13,20)(14,17)(15,18)(16,19)(21,24)(22,23)(25,29,31,35)(26,30,33,39)(27,37,36,28)(32,40,38,34);(1,2)(3,6)(4,5)(7,8)(9,13,14,22)(10,16,15,21)(11,19,18,24)(12,20,17,23)(25,30)(26,35)(27,34)(28,38)(29,33)(31,39)(32,37)(36,40)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=g,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8824932);