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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1009 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)(3,7)(4,5)(6,8)(17,18,21,19)(20,22,24,23);(2,5)(6,7)(9,10)(11,15)(12,13)(14,16)(17,18,21,19)(20,23,24,22);(3,8)(6,7)(10,13)(14,15)(18,19)(22,23);(1,3,4,8)(2,6,5,7)(9,10)(11,14)(12,13)(15,16)(17,19,21,18)(20,23,24,22);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,16)(13,15)(17,20,21,24)(18,22,19,23)Polycyclic presentation:
2^8|a^2=g,b^2=g,d^2=fg,e^2=g,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×#41240, #7638899, #7639598, … |
| maximal normal subgroup of | C2×#41240, #7638899, #7639598, … |
| maximal quotient of | #544902, #545311, #545956, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 34228);