Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| socle | C22 | C23×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1063 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 16 points:
(2,5)(6,8)(10,13)(11,15)(14,16);(1,2)(3,6)(4,5)(7,8)(10,14)(13,16);(9,10)(11,16)(12,14)(13,15);(2,5)(6,8);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^8|e^2=f,g^2=h,b^a=bf,c^a=cg,g^a=gh,c^b=ch,d^b=df,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #221070, #221080, #221090, … |
| maximal normal subgroup of | #221070, #221080, #221090, … |
| maximal quotient of | #6379541, #6379542, #6380377, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54425);