Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
474 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,3)(5,6)(7,11)(8,10)(9,12);(1,2,4,3)(5,7,8,12)(6,9,10,11)(14,17)(15,19)(18,20);(6,10)(9,11)(13,14,16,18)(15,20,19,17);(1,3,4,2)(6,10)(9,11)Polycyclic presentation:
2^8|b^2=e,c^2=h,d^2=ef,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#26928, #6443525, #6443544, … |
| maximal normal subgroup of | C2×#26928, #6443525, #6443544, … |
| maximal quotient of | #312557, #312558, #312565, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 19916);