Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393940 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2408 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3304 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(6,7)(9,10)(11,12);(5,6)(7,8)(13,14)(15,17)(16,19)(18,20);(2,4)(5,7,8,6)(14,17)(15,20)(18,19);(10,12)Polycyclic presentation:
2^9|c^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gi,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72239666, #72239673, #72239789, … |
| maximal normal subgroup of | #72239666, #72239673, #72239789, … |
| maximal quotient of | #72250113, #72322643 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301759);