Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C22×#525 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
7936 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(5,6,7,8)(13,14)(15,19)(16,17)(18,20);(1,2)(3,4)(5,6)(7,8)(10,12)(13,15,17,20)(14,18,16,19);(9,10)(11,12)(13,15)(14,18)(16,19)(17,20);(2,4)(5,6,7,8)(13,15)(14,19)(16,18)(17,20);(1,2)(3,4)(5,6)(7,8)(13,16,17,14)(15,19,20,18)Polycyclic presentation:
2^9|a^2=i,b^2=g,d^2=i,e^2=g,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=df,e^d=ef140 conjugacy classes — view the full character table.
| maximal subgroup of | #72219225, #72225789, #72232421, … |
| maximal normal subgroup of | #72219225, #72219258, #72225789, … |
| maximal quotient of | C2×#519339, #72251003, #72302227, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 426535);