Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C4×C22⋊C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
713 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(10,15)(11,13)(12,19)(16,20)(17,21)(23,24)(25,26,28,31,29,32,35,38)(27,30,34,37,33,36,39,40);(1,3,4,7)(2,5,6,8)(9,10)(11,15)(12,20)(13,21)(14,17)(16,18)(19,23)(22,24)(25,27)(26,30)(28,34)(29,33)(31,37)(32,36)(35,39)(38,40);(2,6)(5,8)(9,11,14,13)(10,15,17,21)(12,18,19,22)(16,23,24,20)(27,33)(30,36)(34,39)(37,40);(9,12,14,19)(10,16,17,24)(11,18,13,22)(15,23,21,20)Polycyclic presentation:
2^9|a^2=g,b^2=fh,c^2=h,d^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,c^b=ci,d^b=de,e^b=eh,e^d=eh104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 132309);