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 | #2464 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
810 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10,14,16)(11,13,12,15)(17,18,21,19)(20,23,24,22);(1,2)(3,5)(4,7)(6,8)(9,11,14,12)(10,15,16,13)(17,19,21,18)(20,23,24,22);(2,3)(6,7)(10,16)(11,12)(17,20,21,24)(18,22,19,23);(1,3,5,2)(4,7,8,6)(9,12,14,11)(10,15,16,13);(1,4)(2,6)(3,7)(5,8)(9,13,14,15)(10,11,16,12)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=fg,d^2=h,e^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×#50730, #7706448, #7706454, … |
| maximal normal subgroup of | C2×#50730, #7706448, #7706454, … |
| maximal quotient of | #560347, #560561, #561541, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 43718);