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) = C42 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #329 |
| commutator | C42 | C42 |
| 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 ▷ C2×C4 ▷ C2 ▷ C1 | |
109 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,2,3,4)(6,10)(7,14)(8,16)(11,19)(12,13)(15,17);(5,6,7,11,8,13,14,20,9,12,15,19,16,10,17,18)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=f,e^2=g,g^2=h,b^a=bc,c^a=cf,e^a=efg,g^a=gh,c^b=cg,f^b=fh,e^c=eh40 conjugacy classes — view the full character table.
| maximal subgroup of | #94680, #94681, #94728, … |
| maximal normal subgroup of | #94680, #94681, #94728, … |
| maximal quotient of | #93378, #93440, #93550, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 254);