Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| 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 | #2682 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
870 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)(9,10,13,14)(11,15,16,12);(2,4)(6,7)(10,14)(12,15)(17,18,20,22)(19,23,24,21);(9,11)(10,12)(13,16)(14,15)(17,18)(19,21)(20,22)(23,24);(1,3,5,8)(2,6,4,7)(9,11)(10,12)(13,16)(14,15);(1,4,5,2)(3,7,8,6)(9,12)(10,11)(13,15)(14,16)(17,19,20,24)(18,21,22,23)Polycyclic presentation:
2^8|a^2=f,b^2=h,d^2=fg,e^2=fgh,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#39325, #7636932, #7637106, … |
| maximal normal subgroup of | C2×#39325, #7636932, #7637106, … |
| maximal quotient of | #521643, #522278, #522919, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32313);