Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | Aut(G) = C24×D4×#120 (order 4096) · Inn(G) = C22×D4 · Out(G) = C24×D4 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | #532 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4⋊C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
355 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,6)(7,8)(9,10,13,18,14,19,23,17)(11,15,12,21,20,16,22,24);(1,2)(3,7)(4,8)(5,6)(9,11,13,12,14,20,23,22)(10,15,18,21,19,16,17,24);(1,3,5,4)(2,7,6,8)(9,12,13,20,14,22,23,11)(10,16,17,21,19,15,18,24);(1,4,5,3)(2,7,6,8)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=ef,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,f^a=fh,c^b=ch,d^b=de58 conjugacy classes — view the full character table.
| maximal subgroup of | #229216, #229217, #229228, … |
| maximal normal subgroup of | #229216, #229217, #229228, … |
| maximal quotient of | #231856, #231857, #231858, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16223);