Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
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 | C22×C22⋊C4 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
713 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6,9,13,10,8,17,12)(7,11,16,19,15,14,20,18);(1,2,3,4)(5,7)(6,11)(8,14)(9,16)(10,15)(12,18)(13,19)(17,20);(5,8)(6,10)(7,14)(9,12)(11,15)(13,17)(16,18)(19,20);(7,15)(11,14)(16,20)(18,19)Polycyclic presentation:
2^8|a^2=g,b^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,f^a=fh,d^b=dh,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #360315, #360316, #360582, … |
| maximal normal subgroup of | #360315, #360316, #360582, … |
| maximal quotient of | #188357, #188358, #188359, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13643);