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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1001 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)(9,10,14,12)(11,15,16,13);(1,3,6,4)(2,7,8,5)(9,11)(10,13)(12,15)(14,16)(17,18,20,22)(19,21,24,23);(2,8)(3,4)(10,12)(11,16)(17,18)(19,23)(20,22)(21,24);(1,4,6,3)(2,5,8,7)(9,12,14,10)(11,15,16,13)(17,19)(18,21)(20,24)(22,23);(1,5)(2,4)(3,8)(6,7)(9,13,14,15)(10,11,12,16)(17,18)(19,23)(20,22)(21,24)Polycyclic presentation:
2^8|a^2=f,b^2=fh,d^2=g,e^2=f,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×#36327, #7629993, #7630075, … |
| maximal normal subgroup of | C2×#36327, #7629993, #7630075, … |
| maximal quotient of | #519174, #519179, #519182, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29315);