Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| 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 | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
861 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,6,4)(3,5,8,7)(9,10)(11,12)(13,15)(14,16)(17,18)(19,20)(21,22)(23,24);(1,3)(2,7)(4,5)(6,8)(11,16)(12,14)(18,22)(20,23);(2,4)(3,8)(9,11,13,16)(10,14,15,12)(17,19)(18,23)(20,22)(21,24);(1,4,6,2)(3,5,8,7)(9,11)(10,12)(13,16)(14,15)(17,20,21,23)(18,19,22,24);(1,5)(2,3)(4,8)(6,7)(9,12)(10,11)(13,14)(15,16)(17,19)(18,23)(20,22)(21,24)Polycyclic presentation:
2^8|a^2=fgh,c^2=fg,d^2=f,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#48845, #7704419, #7704956, … |
| maximal normal subgroup of | C2×#48845, #7704419, #7704956, … |
| maximal quotient of | #560579, #560632, #561680, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 41833);