Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| 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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
762 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,6,4)(3,5,8,7)(9,10,14,13)(11,12,16,15)(17,18,21,19)(20,22,24,23);(1,3,6,8)(2,7,4,5)(11,16)(12,15)(20,24)(22,23);(1,2,6,4)(3,7,8,5)(9,11,14,16)(10,15,13,12)(17,19,21,18)(20,23,24,22);(1,4,6,2)(3,5,8,7)(9,12)(10,11)(13,16)(14,15)(17,20)(18,22)(19,23)(21,24);(1,5)(2,3)(4,8)(6,7)(9,13,14,10)(11,12,16,15)Polycyclic presentation:
2^8|a^2=fh,b^2=f,c^2=fh,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#59732, #7666640, #7666667, … |
| maximal normal subgroup of | C2×#59732, #7666640, #7666667, … |
| maximal quotient of | #532085, #532260, #532288, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 52720);