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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
962 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,3)(4,7,8,6)(17,18,21,23)(19,20,24,22);(2,3)(6,7)(10,13)(11,15)(17,19,21,24)(18,22,23,20);(9,10,12,13)(11,14,15,16)(17,20)(18,19)(21,22)(23,24);(1,3,5,2)(4,7,8,6)(9,10)(11,14)(12,13)(15,16)(17,19)(18,20)(21,24)(22,23);(1,4,5,8)(2,6,3,7)(9,11)(10,14)(12,15)(13,16)(17,20)(18,19)(21,22)(23,24)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=h,d^2=fg,e^2=fg,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×#50063, #7705531, #7705686, … |
| maximal normal subgroup of | C2×#50063, #7705531, #7705686, … |
| maximal quotient of | #560671, #561111, #561776, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 43051);