Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C82 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C42 |
| commutator | C4 | C82 |
| Frattini, cosocle | C2×C4×C8 | C22 |
| socle, prime radical | C23 | #97 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
133 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 28 points:
(1,2)(3,4)(13,14,16,21,17,19,25,26,18,22,24,28,15,20,23,27);(2,4)(5,6,7,9,8,10,11,12)(14,19,22,20)(16,24)(21,27,28,26)(23,25)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=h,d^2=f,e^2=g,f^2=h,b^a=bc,e^a=eh,d^b=dh88 conjugacy classes — view the full character table.
| maximal subgroup of | #92839, #92840, C2×#7137, … |
| maximal normal subgroup of | #92839, #92840, C2×#7137, … |
| maximal quotient of | #92896, #92899, #92900, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 125);