Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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 | #2670 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
857 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,3)(4,6,8,7)(11,15)(14,16)(17,18,20,22)(19,23,24,21);(1,2,5,3)(4,7,8,6)(9,10,13,12)(11,14,15,16)(17,19,20,24)(18,21,22,23);(2,3)(6,7)(9,11)(10,14)(12,16)(13,15)(17,19)(18,21)(20,24)(22,23);(1,3,5,2)(4,7,8,6)(9,10,13,12)(11,14,15,16)(17,18)(19,21)(20,22)(23,24);(1,4,5,8)(2,6,3,7)(9,12,13,10)(11,16,15,14)Polycyclic presentation:
2^8|a^2=h,b^2=g,d^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#41834, #7640071, #7640093, … |
| maximal normal subgroup of | C2×#41834, #7640071, #7640093, … |
| maximal quotient of | #544979, #545338, #545931, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 34822);