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 | #2464 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
884 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,6,5)(3,4,8,7)(9,10)(11,15)(12,16)(13,14)(18,21)(19,23);(3,8)(4,7)(10,14)(11,16)(17,18)(19,22)(20,21)(23,24);(1,3,6,8)(2,7,5,4)(9,11)(10,12)(13,16)(14,15);(1,4)(2,3)(5,8)(6,7)(9,12)(10,11)(13,15)(14,16);(1,5,6,2)(3,4,8,7)(9,12)(10,11)(13,15)(14,16)(17,19,20,23)(18,22,21,24)Polycyclic presentation:
2^8|a^2=h,c^2=h,e^2=fg,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×#59733, #7666749, #7666823, … |
| maximal normal subgroup of | C2×#59733, #7666749, #7666823, … |
| maximal quotient of | #531858, #531940, #532284, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 52721);