Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3045 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
383 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,6)(4,8)(5,7)(9,10)(11,19)(12,16)(13,20)(14,17)(15,18)(21,23)(22,24);(1,3,5,4)(2,6,7,8)(9,11)(10,12)(13,21)(14,16)(15,22)(17,24)(18,23)(19,20);(1,4,5,3)(2,6,7,8)(9,12)(10,16)(11,20)(13,19)(14,24)(15,23)(17,22)(18,21);(9,12)(10,11)(13,24)(14,19)(15,23)(16,20)(17,21)(18,22)Polycyclic presentation:
2^8|b^2=f,c^2=f,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=eh,g^b=gh,g^c=gh,e^d=eh43 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#26190, #6438603, #6438604, … |
| maximal normal subgroup of | C2×#26190, #6438603, #6438604, … |
| maximal quotient of | #306313, #306316, #306325, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 19178);