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) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
841 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6,10,7)(8,15,17,20)(9,11,19,13)(12,18,14,16);(6,11)(7,13)(9,19)(12,15)(14,20)(16,18);(2,4)(5,7)(6,10)(8,14)(9,11)(12,17)(13,19)(15,18)(16,20);(5,8)(6,12)(7,14)(9,18)(10,17)(11,15)(13,20)(16,19)Polycyclic presentation:
2^8|a^2=h,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×#23888, #6437601, #6437602, … |
| maximal normal subgroup of | C2×#23888, #6437601, #6437602, … |
| maximal quotient of | #244500, #244505, #244506, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16876);