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 | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
809 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,4)(3,7,8,6)(10,11)(14,15);(1,3)(2,6)(4,7)(5,8)(10,11)(14,15)(19,23)(21,24);(9,10,13,11)(12,14,16,15)(17,18,20,22)(19,21,23,24);(1,4,5,2)(3,7,8,6)(9,11,13,10)(12,15,16,14)(17,19)(18,21)(20,23)(22,24);(1,3)(2,6)(4,7)(5,8)(9,12,13,16)(10,14,11,15)Polycyclic presentation:
2^8|a^2=f,c^2=h,d^2=fg,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×#59579, #7666782, #7666821, … |
| maximal normal subgroup of | C2×#59579, #7666782, #7666821, … |
| maximal quotient of | #532020, #532071, #532184, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 52567);