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 |
| prime radical | #2407 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1007 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)(11,12)(13,16)(14,15);(1,3)(2,5)(4,7)(6,8)(10,15)(11,13)(17,18,20,22)(19,21,24,23);(2,4)(3,8)(9,11,14,13)(10,12,15,16)(17,18)(19,23)(20,22)(21,24);(1,4,6,2)(3,7,8,5)(9,12)(10,11)(13,15)(14,16)(17,19)(18,21)(20,24)(22,23);(1,5,6,7)(2,3,4,8)(9,13,14,11)(10,16,15,12)(17,18)(19,23)(20,22)(21,24)Polycyclic presentation:
2^8|a^2=g,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 | #164361, #164446, #164449, … |
| maximal normal subgroup of | #164361, #164446, #164449, … |
| maximal quotient of | #519179, #519182, #519246, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29369);