Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| 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 | |
1449 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(5,6)(7,8)(9,10)(11,12)(14,17)(18,20);(6,10)(7,11)(14,17)(15,19);(1,2)(3,4)(13,14)(15,18)(16,17)(19,20);(5,7,9,11)(6,8,10,12)(13,14)(15,18)(16,17)(19,20);(5,8)(6,11)(7,10)(9,12)(13,15)(14,18)(16,19)(17,20)Polycyclic presentation:
2^8|d^2=f,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=dfh,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#45443, #7702041, #7702253, … |
| maximal normal subgroup of | C2×#45443, #7702041, #7702253, … |
| maximal quotient of | #557019, #557028, #557347, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38431);