Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
539 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(5,6,8,7)(11,18)(15,23)(19,24)(20,22);(2,4)(5,7,8,6)(9,10)(11,19)(12,16)(13,17)(14,21)(15,20)(18,24)(22,23);(6,7)(9,11)(10,15)(12,20)(13,18)(14,22)(16,24)(17,23)(19,21);(2,4)(9,12,13,14)(10,16,17,21)(11,20,18,22)(15,24,23,19)Polycyclic presentation:
2^8|a^2=fh,b^2=fh,d^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#25089, #6438746, #6438766, … |
| maximal normal subgroup of | C2×#25089, #6438746, #6438766, … |
| maximal quotient of | #281613, #281614, #281617, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18077);