Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1292 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,3)(5,6,9,8)(7,11,12,10)(13,14,17,20)(15,18,16,19);(1,2,4,3)(5,6,9,8)(7,11,12,10)(13,15,17,16)(14,18,20,19);(1,3,4,2)(5,7)(6,10)(8,11)(9,12)(13,14)(15,18)(16,19)(17,20);(5,7)(6,11)(8,10)(9,12)(13,16,17,15)(14,19,20,18);(5,8,9,6)(7,11,12,10)Polycyclic presentation:
2^8|a^2=fg,b^2=g,c^2=f,d^2=fgh,e^2=h,b^a=bf,c^a=cg,c^b=ch,e^d=eh64 conjugacy classes — view the full character table.
| maximal subgroup of | #132365, #248311, #248312, … |
| maximal normal subgroup of | #132365, #248311, #248312, … |
| maximal quotient of | #517888, #517892, #517906, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28900);