Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | Aut(G) order 65536 |
| 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 | |
1069 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,6,4)(3,7,8,5)(9,10,13,12)(11,14,16,15);(1,3)(2,5)(4,7)(6,8)(10,12)(14,15)(17,18,20,22)(19,21,24,23);(2,4)(3,8)(9,10,13,12)(11,15,16,14)(17,18)(19,23)(20,22)(21,24);(1,4,6,2)(3,7,8,5)(9,11)(10,14)(12,15)(13,16)(17,19)(18,21)(20,24)(22,23);(1,5,6,7)(2,3,4,8)(9,12,13,10)(11,14,16,15)(17,18)(19,23)(20,22)(21,24)Polycyclic presentation:
2^8|a^2=f,b^2=fh,c^2=fg,d^2=g,e^2=f,b^a=bf,c^a=cg,c^b=ch,e^b=eh,d^c=dh,e^d=eh70 conjugacy classes — view the full character table.
| maximal subgroup of | #164347, #164410, #164413, … |
| maximal normal subgroup of | #164347, #164410, #164413, … |
| maximal quotient of | #519205, #519210, #519253, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29357);