Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C2×#2433 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
5410 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)(9,10,12,11)(13,14,17,18)(15,16,20,19);(6,8)(14,18)(16,19);(1,2)(3,4)(5,6)(7,8)(9,11,12,10)(14,18)(15,20);(1,2)(3,4)(13,15)(14,19)(16,18)(17,20);(13,16,17,19)(14,15,18,20)Polycyclic presentation:
2^9|a^2=i,c^2=fi,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,e^b=ef,d^c=df116 conjugacy classes — view the full character table.
| maximal subgroup of | #72261003, #72261652, #72261714, … |
| maximal normal subgroup of | #72261003, #72261652, #72261714, … |
| maximal quotient of | #72298106, #72298404, C2×#536489, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 443685);