Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| 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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
858 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10,14,13)(11,16,12,15)(19,23)(22,24);(1,2,4,6)(3,5,8,7)(9,11,14,12)(10,15,13,16)(17,18,21,20)(19,22,23,24);(1,2)(3,7)(4,6)(5,8)(9,12,14,11)(10,16,13,15)(17,19)(18,22)(20,24)(21,23);(1,3)(2,5)(4,8)(6,7)(9,13,14,10)(11,16,12,15)(17,20,21,18)(19,24,23,22);(1,2)(3,7)(4,6)(5,8)Polycyclic presentation:
2^8|a^2=f,b^2=gh,c^2=f,d^2=g,b^a=bf,c^a=cg,c^b=ch,e^b=eh,d^c=dh,e^d=eh70 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#36343, #7630014, #7630042, … |
| maximal normal subgroup of | C2×#36343, #7630014, #7630042, … |
| maximal quotient of | #519004, #519026, #519073, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29331);