Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1355 |
| automorphism | Aut(G) = #72279281 (order 2048) · Inn(G) = #329 · Out(G) = #120 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #329 |
| commutator | C42 | C2×C8 |
| Frattini, cosocle | C2×#97 | C22 |
| prime radical | C23 | #100 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
103 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10,12,16,15,17,21,23,14,19,22,24,13,18,11,20);(1,3)(2,5)(4,8)(6,7)(9,11,14,21)(10,16)(12,15,22,13)(17,20)(18,23)(19,24)Polycyclic presentation:
2^8|a^2=d,b^2=f,c^2=fh,d^2=g,e^2=h,g^2=h,b^a=bc,c^a=ce,f^a=fh,c^b=cf,d^b=def,e^b=eh,g^b=gh,d^c=dh40 conjugacy classes — view the full character table.
| maximal subgroup of | #94544, #94545, #94600, … |
| maximal normal subgroup of | #94544, #94545, #94600, … |
| maximal quotient of | #92885, #92892, #92893, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 118);