Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | Aut(G) = C24×D4×#120 (order 4096) · Inn(G) = C22×D4 · Out(G) = C27 |
| 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 | C42 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | #3106 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
346 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,6)(3,4)(7,8)(9,10,12,16,13,15,19,22)(11,14,17,21,18,20,23,24);(1,2)(3,6)(4,8)(5,7);(1,3,5,4)(2,6,7,8)(9,11,12,17,13,18,19,23)(10,14,16,21,15,20,22,24);(1,3,5,4)(2,6,7,8)(10,15)(14,20)(16,22)(21,24)Polycyclic presentation:
2^8|a^2=f,c^2=fg,d^2=g,e^2=g,f^2=h,b^a=be,c^a=cg,d^a=dgh,e^a=eg,e^b=eg64 conjugacy classes — view the full character table.
| maximal subgroup of | #136761, #136762, #137866, … |
| maximal normal subgroup of | #136761, #136762, #137866, … |
| maximal quotient of | #182503, #182537, #183454, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10905);