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 | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
908 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10)(11,12)(13,15)(14,16)(17,18,21,24)(19,22,20,23);(1,2,4,6)(3,5,8,7)(9,11,14,13)(10,12,16,15)(17,18)(19,22)(20,23)(21,24);(1,2)(3,7)(4,6)(5,8)(9,12,14,15)(10,13,16,11)(17,18)(19,22)(20,23)(21,24);(1,3)(2,5)(4,8)(6,7)(9,13,14,11)(10,15,16,12)(17,19,21,20)(18,22,24,23);(1,2)(3,7)(4,6)(5,8)(9,10)(11,15)(12,13)(14,16)(17,20,21,19)(18,23,24,22)Polycyclic presentation:
2^8|a^2=f,b^2=h,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 | #164147, #164229, #164232, … |
| maximal normal subgroup of | #164147, #164229, #164232, … |
| maximal quotient of | #519203, #519221, #519253, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29355);