Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| 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 |
| prime radical | #2555 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
879 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,5)(6,8)(10,15)(11,16)(18,22)(23,24);(2,5)(6,7)(9,10,14,15)(11,12,16,13)(17,18)(19,23)(20,24)(21,22);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,15)(14,16)(17,19,21,20)(18,23,22,24);(1,2)(3,6)(4,5)(7,8)(9,12,14,13)(10,16,15,11);(1,3,4,8)(2,6,5,7)(9,13,14,12)(10,11,15,16)(17,20,21,19)(18,24,22,23)Polycyclic presentation:
2^8|b^2=g,c^2=fgh,d^2=g,e^2=f,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#35127, #7628338, #7628361, … |
| maximal normal subgroup of | C2×#35127, #7628338, #7628361, … |
| maximal quotient of | #516521, #516553, #516686, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28115);