Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4⋊C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
429 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6,9,13,10,7,14,12)(8,11,18,20,19,15,17,16);(5,7)(6,10)(8,16)(9,12)(11,18)(13,14)(15,17)(19,20);(1,2,3,4);(5,8)(6,11)(7,15)(9,18)(10,19)(12,16)(13,20)(14,17)Polycyclic presentation:
2^8|a^2=g,c^2=f,e^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,d^b=de,e^b=eh,e^d=eh58 conjugacy classes — view the full character table.
| maximal subgroup of | #136344, #136371, #136400, … |
| maximal normal subgroup of | #136344, #136371, #136400, … |
| maximal quotient of | #225430, #225431, #225432, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15674);