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 | |
864 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,7)(5,8)(10,14)(11,16)(17,18)(19,23)(20,24)(21,22);(1,2)(3,5)(4,8)(6,7)(10,14)(11,16)(18,22)(19,24);(1,3,6,4)(2,5,7,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,20)(21,24)(22,23);(1,4,6,3)(2,8,7,5)(9,11,13,16)(10,15,14,12)(17,20)(18,19)(21,23)(22,24);(1,5,6,8)(2,3,7,4)(9,12,13,15)(10,16,14,11)(17,20)(18,19)(21,23)(22,24)Polycyclic presentation:
2^8|c^2=fgh,d^2=fg,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×#59619, #7666891, #7666900, … |
| maximal normal subgroup of | C2×#59619, #7666891, #7666900, … |
| maximal quotient of | #532010, #532301, #532303, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 52607);