Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24269 |
| 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 | C2×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
493 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(5,6,8,7)(9,10)(11,16)(12,21)(13,23)(14,18)(15,20)(17,22)(19,24);(1,2)(3,4)(5,7,8,6)(9,11,13,20,14,19,22,12)(10,15,17,16,18,21,23,24);(2,4)(6,7)(11,19)(12,20)(15,21)(16,24);(9,12,22,19,14,20,13,11)(10,16,23,15,18,24,17,21)Polycyclic presentation:
2^8|a^2=f,b^2=ef,d^2=eh,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^c=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#25418, #6444220, #6444235, … |
| maximal normal subgroup of | C2×#25418, #6444220, #6444235, … |
| maximal quotient of | #288109, #288113, #288117, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18406);