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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
795 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(9,10,13,14)(11,15,16,12)(17,18,22,23)(19,26,24,20)(21,25,29,30)(27,32,31,28);(2,6)(4,7)(11,16)(12,15)(18,23)(19,24)(25,30)(27,31);(1,2)(3,7)(4,8)(5,6)(9,11)(10,12)(13,16)(14,15)(17,19,22,24)(18,20,23,26)(21,27,29,31)(25,28,30,32);(1,3,5,8)(2,7,6,4)(17,20)(18,24)(19,23)(21,28)(22,26)(25,31)(27,30)(29,32);(1,4,5,7)(2,8,6,3)(9,12)(10,11)(13,15)(14,16)(17,21,22,29)(18,25,23,30)(19,27,24,31)(20,28,26,32)Polycyclic presentation:
2^8|a^2=g,c^2=f,d^2=gh,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×#40408, #7637791, #7638138, … |
| maximal normal subgroup of | C2×#40408, #7637791, #7638138, … |
| maximal quotient of | #522015, #522537, #523121, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 33396);