Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35346 |
| 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×#120 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
912 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,5)(4,7)(6,8)(9,10,14,16)(11,13,15,12)(17,18,21,24)(19,22,20,23);(1,3,6,4)(2,7,8,5)(9,11,14,15)(10,12,16,13)(17,18)(19,22)(20,23)(21,24);(2,8)(3,4)(9,10)(11,12)(13,15)(14,16);(1,4,6,3)(2,7,8,5)(9,12,14,13)(10,11,16,15)(17,19,21,20)(18,22,24,23);(1,5,6,7)(2,3,8,4)(9,13,14,12)(10,15,16,11)(17,20,21,19)(18,23,24,22)Polycyclic presentation:
2^8|a^2=fh,b^2=g,d^2=f,e^2=f,b^a=bf,c^a=cg,c^b=ch,e^b=eh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#35462, #7628831, #7628855, … |
| maximal normal subgroup of | C2×#35462, #7628831, #7628855, … |
| maximal quotient of | #515321, #515345, #515480, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28450);