Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
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 | |
1020 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10,15,16)(11,12,14,13)(17,18)(19,20)(21,23)(22,24);(1,2)(3,6)(4,7)(5,8)(9,11,15,14)(10,13,16,12)(17,18)(19,20)(21,23)(22,24);(1,3)(2,5)(4,8)(6,7)(9,12,15,13)(10,11,16,14)(17,19)(18,22)(20,23)(21,24);(1,3)(2,6)(4,8)(5,7)(9,13,15,12)(10,11,16,14)(17,20,21,22)(18,19,23,24);(1,2)(3,5)(4,7)(6,8)(9,14,15,11)(10,13,16,12)(17,19)(18,20)(21,24)(22,23)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=f,d^2=g,e^2=f,b^a=bf,c^a=cg,c^b=ch,e^d=eh64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#35744, #7629314, #7629399, … |
| maximal normal subgroup of | C2×#35744, #7629314, #7629399, … |
| maximal quotient of | #517716, #517718, #517719, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28732);